Proofs from THE BOOK: verified results and open formalization tasks

This Textbook project develops a reusable Lean library around Martin Aigner and Günter M. Ziegler's Proofs from THE BOOK. It combines results imported from the existing proof_in_the_book repository with precise contribution targets from the sixth edition (2018). The aim is to preserve mathematical meaning, reuse existing proofs, and make the remaining work accessible to other contributors.

What is already verified

The original import contains 156 distinct platform-accepted results. Euclid, the original main theorem, is retained as a completed milestone when the project goal moves to the sixth-edition extension. Every one of the repository's 40 chapter topics has accepted results. Each result certifies its actual Lean statement, including its hypotheses; this does not certify every argument or every theorem in a chapter. Some proofs reuse Mathlib, while others were developed in the repository. Their source and proof notes retain that distinction.

The imported source snapshot is 873d52e0c88cd351f594221e70c3c5b3559777a9. Imported results use Lean 4.30.0 and Mathlib c5ea00351c28e24afc9f0f84379aa41082b1188f. Immutable public source links are used only where the linked source matches the verified artifact. Compatibility changes, unsuccessful attempts, and verification evidence are retained in the integration project.

The live goal is the explicit conjunction of the 21 linked sixth-edition extension targets. Its reduction connects these targets to the goal, so proving the remaining children advances the project. This goal is deliberately narrower than “every theorem and every proof in the book”; the unlinked topology tasks below are additional formalization work.

Sixth-edition contribution targets

New milestones explicitly marked 6th ed. cover Chapters 7 (spectral theorem and determinants), 15 (round circles and links), 35 (finite Kakeya), 37 (permanents and entropy), and 45 (probabilistic counting). They include the precise definitions and boundary conditions needed to state the results. Compiled Open targets are requests for proofs, not proved results. The spectral theorem has a direct Mathlib proof; community results are reused under their actual statements and with attribution.

Two Chapter 15 tasks intentionally remain unlinked mathematical milestones: the full non-equivalence assertion for the depicted Borromean, Tait, and trivial links, and the Fox-coloring invariance bridge for equivalent link diagrams. These invite formalization of the diagrams and the topology bridge as well as proof. The separate modular Fox calculations do not by themselves establish ambient non-equivalence.

The crossing-lemma target is the universal good-drawing form: actual injective edge arcs and exact finite intersection records appear in its interface. It does not assume the desired crossing bound. The Ramsey target preserves the real exponent for odd k. The related public Erdős–Ramsey result with a rounded exponent is identified as a supporting result, not as proof of that full target.

Chapter numbering and statement scope

Older milestones use the repository's chapter labels. Repository Chapters 1–21 match the bundled fourth edition; Chapter 22 inserts Van der Waerden's permanent theorem, and Chapters 23–40 correspond to fourth-edition Chapters 22–39. The sixth edition has 45 chapters, so these organizational labels are not sixth-edition chapter numbers. New milestones give sixth-edition numbers and printed source pages explicitly.

Some existing formalizations preserve narrower statements or additional premises. Examples include repository Chapter 13's dihedral-angle conclusion, Chapter 28's Dilworth lower-bound result, and the geometric premises in Chapter 36. Read the actual linked theorem and its description before reusing it. A chapter title or the former Euclid main theorem is not a completion certificate for the collection.

How to contribute

Choose an Open linked theorem and inspect its definitions, exact binders, Mathlib revision, and prior attempts. Reuse a compatible existing result when it proves that statement; preserve the original contributor's attribution. Submit a matching proof for verification. For an unlinked milestone, first formalize and review the source statement and its definitions. These are known textbook results awaiting formalization or proof in this project, rather than claims of new unresolved mathematics.

Source repository: https://github.com/xiangyazi24/proof_in_the_book

Book: Aigner and Ziegler, Proofs from THE BOOK, Sixth Edition (2018), https://doi.org/10.1007/978-3-662-57265-8

Motivation

Machine scheduling asks how to assign jobs to machines over time. In many applications a job also needs additional scarce resources while it runs: a tool, a skilled operator, a memory bank, a channel. Adding such resources can turn a problem with a polynomial algorithm into an NP-hard one. Błażewicz, Lenstra and Rinnooy Kan (DAM 1983) extended the standard three-field classification $\alpha\,|\,\beta\,|\,\gamma$ of scheduling problems (Graham, Lawler, Lenstra and Rinnooy Kan 1979) with a resource field $res\lambda\sigma\rho$. They then drew the complete borderline between easy and hard problems for unit-time jobs, identical or uniform machines and the makespan criterion. Their Fig. 2 marks each problem type as polynomially solvable or NP-hard.

This mission formalizes the two hardness results of that classification that come from graph partition problems (Theorems 2 and 3, p. 15). Two identical machines are easy under any resource constraints (Theorem 1, after Garey and Johnson 1975). Theorems 2 and 3 show that a third identical machine, or two machines of different speeds, already makes the problem strongly NP-hard, once the number of unit resources is part of the input.

Setting

There are $n$ jobs $J_1,\dots,J_n$ and $m$ machines $M_1,\dots,M_m$. Each machine processes at most one job at a time, and each job runs on one machine without interruption. Machine $M_i$ has a speed $q_i>0$, and every job has unit execution requirement $p_j=1$, so it takes time $1/q_i$ on $M_i$. Identical machines ($P$) are the case $q_i=1$; uniform machines ($Q$) allow arbitrary speeds.

There are $l$ resources $R_1,\dots,R_l$. Resource $R_h$ has a positive integer size $s_h$, the amount available at any time. Job $J_j$ needs a nonnegative integer amount $r_{hj}$ of $R_h$ throughout its execution. A schedule assigns each job a machine $\mu(j)$ and a start time $S_j\ge 0$. Its completion time is $C_j=S_j+1/q_{\mu(j)}$, and it is being executed at every time $t$ with $S_j\le t<C_j$. A schedule is feasible when:

• jobs on the same machine do not overlap in time;
• at every time $t$, the set $S_t$ of jobs being executed satisfies

The makespan is $C_{\max}=\max_j C_j$.

The resource type $res{\cdot}11$ means three things: the number $l$ of resources is part of the input, every size is $s_h=1$, and every requirement satisfies $r_{hj}\le1$. A unit resource is therefore a conflict: two jobs that both need it can never run at the same time. The problems here have no precedence constraints. $Pm\,|\,res{\cdot}11,\,p_j=1\,|\,C_{\max}$ and $Qm\,|\,res{\cdot}11,\,p_j=1\,|\,C_{\max}$ ask for a feasible schedule of minimum makespan. Their decision versions ask, for a threshold $y$, whether a feasible schedule with $C_{\max}\le y$ exists.

The source problems are two graph problems on a graph $G=(V,E)$ with $|V|=3t$:

• PARTITION INTO TRIANGLES: can $V$ be partitioned into $t$ triples of pairwise adjacent vertices?
• PARTITION INTO PATHS OF LENGTH 2: can $V$ be partitioned into $t$ triples, each with at most one nonadjacent pair, that is, each spanning a path of length 2?

Both are NP-complete (Garey and Johnson 1979, problems GT11 and GT13).

The construction of p. 15 introduces one job per vertex and one unit resource $R_{\{j,k\}}$ per nonadjacent pair $\{j,k\}$, required by $J_j$ and $J_k$ only.

Formalization targets

Goal: Theorem 3

Formally: if the language of PARTITION INTO PATHS OF LENGTH 2 is NP-hard, then the language of unary codes of yes-instances of the decision version of $Q2\,|\,res{\cdot}11,\,p_j=1\,|\,C_{\max}$ is NP-hard. The two speeds are arbitrary positive integers.

Milestones

1. The construction's key property (p. 15). In the constructed instance, two distinct jobs can be executed simultaneously if and only if their vertices are adjacent.
2. The triangle equivalence (proof of Theorem 2). $G$ has a partition into triangles if and only if the constructed instance on three identical machines has a feasible schedule with $C_{\max}\le t$.
3. Theorem 2. $P3\,|\,res{\cdot}11,\,p_j=1\,|\,C_{\max}$ is NP-hard in the strong sense, given the NP-hardness of PARTITION INTO TRIANGLES.
4. The paths equivalence (proof of Theorem 3). $G$ has a partition into paths of length 2 if and only if the constructed instance on two uniform machines with speeds $q_1=2$, $q_2=1$ has a feasible schedule with $C_{\max}\le t$.

Significance

The results. Theorems 2 and 3 are two of the minimal NP-hard problems in the paper's classification. Together with Theorem 1 they place the borderline exactly: with unit resources whose number is part of the input, two identical machines are polynomial, while three identical machines, or two machines of different speeds, are strongly NP-hard. Strong NP-hardness rules out pseudo-polynomial algorithms unless P = NP, and it carries over to every more general resource type and machine environment in Fig. 1 and Fig. 2. Section 4.1 of the paper also derives hardness for $\sum C_j$ and $L_{\max}$ from these instances.

Formalizing it. The results are classical and proved on paper, but the paper's proofs are one sentence each ("Clearly", "It is easily seen"). No machine-checked proof exists, and the platform has no model of resource-constrained scheduling with real-valued time. This mission produces that model. It also produces a precise statement of strong NP-hardness on top of Cook's Turing-machine definitions, and the first formal NP-hardness reductions from graph partition problems to scheduling.

Difficulty

The scheduling half of each equivalence depends on the real-time model. On two uniform machines of speeds 2 and 1, jobs take time $\tfrac12$ and $1$, so jobs on the fast machine start at half-integers or anywhere else. The resource constraint must hold at every real time, not at a finite set of checkpoints. An argument that treats time as integer slots applies to the triangle case but does not transfer to the paths case.

The complexity half needs polynomial-time computability of the construction on Cook's one-tape Turing machines, on encoded strings that include malformed inputs. It also needs closure of polynomial-time reductions under composition, which the imported complexity layer states but does not prove.

Formalization scope

• Time and schedules. Start times are nonnegative reals, execution intervals are half-open $[S_j,C_j)$, and the resource constraints are imposed at every real time. Schedules are nonpreemptive.
• Indices. Jobs, machines and resources are 0-based (Fin n, Fin m, Fin l), so $q_1,q_2$ are q 0, q 1.
• Decision versions. "NP-hard" refers to the decision version with a threshold $y$. Thresholds are natural numbers and the $Q2$ speeds are positive integers. This restricted problem is a subproblem of the one with rational data, so its hardness is the stronger statement.
• Encodings and strong NP-hardness. Instances are strings over a two-letter alphabet with every number in unary. Graphs are $t$ in unary followed by the $3t\times 3t$ adjacency matrix, so $|V|=3t$ is part of the instance. Languages contain only codes of yes-instances. Strong NP-hardness is NP-hardness of the unary code language. With unary numbers, $\mathrm{Max}(I)\le\mathrm{Length}(I)$, so this is equivalent to Garey and Johnson's definition. The complexity layer is the published module CookPvsNP_defs.
• Cited hypothesis. Each hardness theorem takes as its only hypothesis the NP-hardness of its source problem, which the paper cites from Garey and Johnson rather than proves. The hypothesis is a true statement about a nonempty, non-universal language. The statements are not weakened to a reduction between languages, and they assume nothing about P versus NP.
• Source problems. The paper's phrase "three vertices, at most two of which are nonadjacent" is read as "at most one nonadjacent pair", which is Garey and Johnson's GT13. Reading it as "at most two nonadjacent pairs" would admit triples with a single edge and change the problem. PARTITION INTO PATHS OF LENGTH 2 reuses the published definition CubicP3Partition.P3Factor, a spanning non-induced $P_3$-factor.
• Construction. Resources are indexed by the nonadjacent pairs $j<k$ in lexicographic order, one per unordered pair and none for a pair $\{j,j\}$. A diagonal resource would make every job infeasible.
• Not trivial. A model that checks resources only at integer times, or only at start times, would make the paths equivalence false. A hypothesis on the target problem would make the goal circular. The definitions rule out both.

Welcome contributions: proofs of the two equivalences, polynomial-time computability of the construction on Cook's machines, and a general composition lemma for polynomial-time reductions. The composition lemma is reusable for every hardness mission built on CookPvsNP_defs.

Selected references

• J. Błażewicz, J. K. Lenstra, A. H. G. Rinnooy Kan, Scheduling subject to resource constraints: classification and complexity, Discrete Applied Mathematics 5 (1983) 11–24. https://doi.org/10.1016/0166-218X(83)90012-4
• M. R. Garey, D. S. Johnson, Complexity results for multiprocessor scheduling under resource constraints, SIAM Journal on Computing 4 (1975) 397–411. https://doi.org/10.1137/0204035
• M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979, ISBN 0-7167-1045-5.
• R. L. Graham, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Annals of Discrete Mathematics 5 (1979) 287–326. https://doi.org/10.1016/S0167-5060(08)70356-X
• S. Cook, The P versus NP problem, Clay Mathematics Institute Millennium Problems. https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf

Motivation

Several classical min-max theorems of combinatorial optimization say that a discrete maximization problem and a continuous minimization problem have the same optimal value, and that both have integral optimal solutions when the data are integral. Edmonds' polymatroid intersection theorem (1970), Fujishige's Fenchel-type duality for submodular functions (1984), Frank's discrete separation theorem for a submodular/supermodular pair (1982), and the potential characterizations of weighted matroid intersection (Frank's weight splitting theorem, 1981; Iri and Tomizawa's criterion for the assignment problem, 1976) are instances. Murota's paper Convexity and Steinitz's exchange property, 1996 places all of them under one theorem: a Fenchel-type min-max formula for a pair of an M-concave and an M-convex function, with integrality on both sides.

Timeline:

• 1970: Edmonds proves the polymatroid intersection theorem.
• 1982: Frank proves the discrete separation theorem for submodular/supermodular set functions, with integrality.
• 1984: Fujishige proves a Fenchel-type min-max theorem for submodular functions.
• 1976–1981: Iri and Tomizawa characterize optimality for independent assignment by potentials; Frank proves the weight splitting theorem for weighted matroid intersection (1981).
• Early 1990s: Dress and Wenzel introduce valuated matroids.
• 1995–1996: Murota proves the valuated matroid intersection theorem (SIAM J. Discrete Math. 9, 1996) and the M-concave intersection theorem (Bonn report, 1995), and in the present paper the Fenchel-type duality (Theorem 6.4).
• Later: the result becomes the central duality theorem of discrete convex analysis (Murota, Discrete Convex Analysis, SIAM, 2003).

Setting

Let $V$ be a finite nonempty set. For $u\in V$, $\chi_u\in\mathbb Z^V$ is its characteristic vector; for $x\in\mathbb R^V$, $\operatorname{supp}^{\pm}(x)$ are the sets of coordinates where $x$ is positive or negative, $x(X)=\sum_{v\in X}x(v)$, and $\langle p,x\rangle=\sum_v p(v)x(v)$.

A finite integral base set is a finite nonempty $B\subseteq\mathbb Z^V$ such that for $x,y\in B$ and $u\in\operatorname{supp}^+(x-y)$ some $v\in\operatorname{supp}^-(x-y)$ has $x-\chi_u+\chi_v\in B$. These are exactly the integer points of integral base polytopes of submodular systems. $\overline B$ is the convex hull of $B$.

A function $\omega:B\to\mathbb R$ has the exchange property (EXC), and is called M-concave, if for $x,y\in B$ and $u\in\operatorname{supp}^+(x-y)$ some $v\in\operatorname{supp}^-(x-y)$ has $x-\chi_u+\chi_v,\ y+\chi_u-\chi_v\in B$ and

A function $\zeta$ is M-convex when $-\zeta$ is M-concave.

For $\omega:B_1\to\mathbb R$ and $\zeta:B_2\to\mathbb R$ the concave conjugate and convex conjugate are

and the concave closure and convex closure are $\hat\omega(b)=\inf_p(\langle p,b\rangle-\omega^\circ(p))$ and $\check\zeta(b)=\sup_p(\langle p,b\rangle-\zeta^\bullet(p))$; they are finite exactly on $\overline{B_1}$ and $\overline{B_2}$.

The primal problem maximizes $\omega(x)-\zeta(x)$ over $x\in B_1\cap B_2$; the relaxed primal problem maximizes $\hat\omega(b)-\check\zeta(b)$ over $b\in\overline{B_1}\cap\overline{B_2}$; the dual problem minimizes $\zeta^\bullet(p)-\omega^\circ(p)$ over $p\in\mathbb R^V$. A maximum over an empty family is $-\infty$.

Formalization targets

Goal: Theorem 6.4

If $\omega$ and $-\zeta$ satisfy (EXC), then

with (P1) a finite dual infimum forces $B_1\cap B_2\neq\emptyset$, and (P2) if $B_1\cap B_2\neq\emptyset$ all values are finite and equal and the infimum is attained. If $\omega,\zeta$ are integer-valued, the infimum may be taken over $p\in\mathbb Z^V$ and is attained there when finite.

Milestones

1. Lemma 6.3 (weak duality): for arbitrary $\omega,\zeta$ on finite nonempty sets, primal $\le$ relaxed $=$ dual (the Fenchel identity (6.5)).
2. Lemma 6.1: $(-f)^\circ(p)=-f^\bullet(-p)$ and $(-f)^\wedge=-\check f$ on $\overline B$.
3. Lemma 4.5: an M-concave $\omega$ satisfies $\hat\omega=\omega$ on $B$.
4. Theorem 2.1: (B1) is equivalent to being the integer points of an integral submodular (or supermodular) base polytope, with the describing functions $\max_{x\in B}x(X)$ and $\min_{x\in B}x(X)$.
5. Theorem 6.5 (Frank's discrete separation theorem, cited in the paper).
6. Lemma 6.7: four equivalent forms of boundedness of the dual problem.
7. Theorem 6.6 (the M-concave intersection theorem, cited in the paper): optimality of $x^*$ for $\omega_1+\omega_2$ is equivalent to a potential $p^*$ with $x^*$ maximizing both $\omega_1[-p^*]$ and $\omega_2[p^*]$, integral when the data are.

Significance

The formula gives, in one statement, the integrality of an optimal solution of the relaxed primal problem (the essential content of the first half, as the paper observes on p. 296) and of the dual problem. The paper presents it as a unification of two groups of theorems: Edmonds' polymatroid intersection theorem, Fujishige's Fenchel-type duality and Frank's discrete separation theorem on one side, and Iri and Tomizawa's potential characterization for independent assignment with its extensions by Fujishige and Frank (weight splitting) on the other. In the paper it yields the primal and dual separation theorems (Theorems 6.8, 6.9) and the convolution results (Theorems 6.10, 6.11), and it is the prototype of the Fenchel-type duality of discrete convex analysis.

All results here are proved in the literature; none is known to be formalized. Mathlib has no submodular base polytopes, no matroid intersection theorem and no discrete convex analysis. A formal proof of Theorem 6.4 would also require formal proofs of the two cited results, Frank's discrete separation theorem and the M-concave intersection theorem, which the paper uses without proof.

Difficulty

Lemma 6.3 is polyhedral convex duality and holds for any functions. The content is equality with the integral problem: the relaxed maximum over the polytope $\overline{B_1}\cap\overline{B_2}$ must be attained at an integer point. For general finite sets it is not, and the intersection of two integral polytopes generally has fractional vertices. Both the integrality of $\overline{B_1}\cap\overline{B_2}$ (Edmonds) and the existence of an integral optimal potential depend on the exchange structure; a direct argument from the definitions of conjugates does not see it. The dual integrality claim, that $p$ can be taken integral, is again specific to (EXC) and fails for general concave extensions.

Formalization scope

Lean conventions, all in namespace SteinitzExchange.Duality:

• $V$ is a type with [Fintype V] [DecidableEq V] [Nonempty V]; integer vectors are V → ℤ, real vectors V → ℝ; finite sets of integer vectors are Finset (V → ℤ).
• A function on $B$ is a total (V → ℤ) → ℝ used only at points of $B$. M-convexity of $\zeta$ is (EXC) for fun x => -ζ x; $\omega$ lives on $B_1$ and $\zeta$ on $B_2$, which are distinct sets in general.
• Conjugates are real-valued min/max over the finite set. The closures are real ⨅/⨆ over $p\in\mathbb R^V$ and are evaluated only on the convex hulls, where they equal the paper's values; off the hulls they carry a junk value instead of $\mp\infty$, which no statement uses.
• The three optimal values are in EReal, as suprema and infima of coerced reals, so no $\infty-\infty$ occurs. EReal's supremum of the empty family is $-\infty$, the paper's convention. The dual infimum is never a real ⨅ (which would return $0$ when unbounded and make (P1) meaningless).
• Every "max" of the page includes attainment: (P2) asserts points $x$, $b$, $p$ at which the three values are achieved; the integral dual infimum is attained when it is not $-\infty$.
• "Integer-valued" means $\omega(x)\in\mathbb Z$ on $B_1$ and $\zeta(x)\in\mathbb Z$ on $B_2$; integral potentials and separating vectors are V → ℤ.
• Theorem 2.1's "$\forall X\subset V$" is read as all $X\subseteq V$.

Formalizations that would trivialize the goal are excluded: an unrestricted real infimum for the dual, a convex closure built from $\zeta^\circ$ instead of $\zeta^\bullet$, a single base set for both functions, and a relaxed maximum taken over all of $\mathbb R^V$ instead of $\overline{B_1}\cap\overline{B_2}$.

Needed infrastructure: finite convex hulls and polyhedral Fenchel duality, submodular base polytopes and their integrality, Frank's separation theorem, and the valuated intersection theorem. The submodular-system layer (Theorem 2.1, Theorem 6.5) is reusable beyond this mission. Contributions to any milestone, including proofs of the two cited theorems, are welcome.

Selected references

• K. Murota, Convexity and Steinitz's exchange property, Advances in Mathematics 124 (1996) 272–311. https://doi.org/10.1006/aima.1996.0084
• K. Murota, Valuated matroid intersection I: optimality criteria, SIAM J. Discrete Math. 9 (1996) 545–561.
• K. Murota, Submodular flow problem with a nonseparable cost function, Report 95843-OR, Forschungsinstitut für Diskrete Mathematik, Universität Bonn, 1995 (source of Theorem 6.6).
• A. Frank, An algorithm for submodular functions on graphs, Annals of Discrete Mathematics 16 (1982) 97–120 (source of Theorem 6.5).
• A. Frank, A weighted matroid intersection algorithm, J. Algorithms 2 (1981) 328–336.
• J. Edmonds, Submodular functions, matroids and certain polyhedra, in: Combinatorial Structures and Their Applications, Gordon and Breach, New York, 1970, 69–87.
• S. Fujishige, Theory of submodular programs: a Fenchel-type min-max theorem and subgradients of submodular functions, Mathematical Programming 29 (1984) 142–155.
• M. Iri and N. Tomizawa, An algorithm for finding an optimal "independent assignment", J. Oper. Res. Soc. Japan 19 (1976) 32–57.
• K. Murota, Discrete Convex Analysis, SIAM, 2003. https://doi.org/10.1137/1.9780898718508

Motivation

Linear optimization over the bases of a matroid, over the integer points of a polymatroid, or over the flows of a network is well understood: the greedy algorithm is exact, the feasible sets are the integer points of polytopes described by submodular functions, and min-max theorems of Edmonds and Frank hold with integrality. Nonlinear objectives on the same sets are much less uniform. Valuated matroids (Dress and Wenzel, 1990; see Murota 2003) showed that a quantitative form of the Steinitz exchange axiom is exactly what keeps the greedy algorithm exact for a nonlinear weight. Kazuo Murota's paper Convexity and Steinitz's Exchange Property (Adv. Math. 124 (1996) 272–311) extends this exchange axiom from matroid bases to the integer points of arbitrary integral base polytopes, names the resulting functions M-concave, and proves that they are the discrete counterpart of concave functions. The paper is the starting point of discrete convex analysis (Murota, Discrete Convex Analysis, SIAM 2003), which is now used in auction theory (gross-substitutes valuations are M♮-concave), inventory and resource allocation, and combinatorial optimization.

This mission covers the first of the paper's three characterizations of M-concavity: the Extension Theorem (Theorem 4.6).

Setting

Let $V$ be a finite nonempty set. For $u\in V$, $\chi_u\in\mathbb Z^V$ is the characteristic vector of $u$. For $x\in\mathbb R^V$, $\operatorname{supp}^+(x)=\{v\mid x(v)>0\}$, $\operatorname{supp}^-(x)=\{v\mid x(v)<0\}$, $\|x\|=\sum_v|x(v)|$, and $\langle p,x\rangle=\sum_v p(v)x(v)$.

A finite integral base set is a finite nonempty $B\subseteq\mathbb Z^V$ such that for $x,y\in B$ and $u\in\operatorname{supp}^+(x-y)$ there is $v\in\operatorname{supp}^-(x-y)$ with $x-\chi_u+\chi_v\in B$ (axiom (B1)). Examples are the incidence vectors of the bases of a matroid. Its convex hull $\overline B$ is an integral base polytope; in general, an integral base polytope is the convex hull of some finite integral base set.

A function $\omega:B\to\mathbb R$ satisfies the exchange property (EXC), and is called M-concave, if for all $x,y\in B$ and $u\in\operatorname{supp}^+(x-y)$ there is $v\in\operatorname{supp}^-(x-y)$ with $x-\chi_u+\chi_v\in B$, $y+\chi_u-\chi_v\in B$ and

The local exchange property (EXC$_{\mathrm{loc}}$) asks only, for $x,y\in B$ with $\|x-y\|=4$, for some $u\in\operatorname{supp}^+(x-y)$ and some $v\in\operatorname{supp}^-(x-y)$ with the same conclusion.

For $p\in\mathbb R^V$, $\omega[p](x)=\omega(x)+\langle p,x\rangle$, and $\operatorname{argmax}(\omega)=\{x\in B\mid\omega(x)\ge\omega(y)\ \forall y\in B\}$. For any $g:B\to\mathbb R$, the concave conjugate is $g^\circ(p)=\min_{x\in B}(\langle p,x\rangle-g(x))$ and the concave closure is $\hat g(b)=\inf_{p\in\mathbb R^V}(\langle p,b\rangle-g^\circ(p))$, a concave function that is finite exactly on $\overline B$. A function $\bar\omega:\overline B\to\mathbb R$ extends $\omega$ if $\bar\omega=\omega$ on $B$.

Formalization targets

Goal: the Extension Theorem (Theorem 4.6)

For a finite integral base set $B$ and $\omega:B\to\mathbb R$,

Milestones

• Lemma 3.2 (p. 282): under (EXC$_{\mathrm{loc}}$), for $y=x-\chi_{u_0}-\chi_{u_1}+\chi_{v_0}+\chi_{v_1}\in B$, $\omega_p(y)-\omega_p(x)\le\max(\pi_{00}+\pi_{11},\pi_{01}+\pi_{10})$ with $\pi_{ij}=\omega_p(x-\chi_{u_i}+\chi_{v_j})-\omega_p(x)$ ($-\infty$ off $B$).
• Theorem 3.1 (p. 282): (EXC) $\iff$ (EXC$_{\mathrm{loc}}$).
• Theorem 2.2 (p. 280): (EXC) for $\omega$ implies (EXC) for every $\omega[p]$.
• Lemma 4.3 (p. 285): under (EXC), $\operatorname{argmax}(\omega)$ is an integral base set.
• Lemma 4.1 (p. 285): $\hat g\ge g$ on $B$; $\max_{\overline B}\hat g=\max_B g$; $\operatorname{argmax}(\hat g)=\overline{\operatorname{argmax}(g)}$.
• Lemma 4.2 (p. 285): $(g[p_0])^\circ(p)=g^\circ(p-p_0)$ and $(g[p_0])^\wedge=\hat g+\langle p_0,\cdot\rangle$ on $\overline B$.
• Theorem 4.4 (p. 286): (EXC) $\iff$ $\operatorname{argmax}(\omega[p])$ is an integral base set for every $p$.
• Lemma 4.5 (p. 288): under (EXC), $\hat\omega=\omega$ on $B$.

Significance

The Extension Theorem identifies a combinatorial axiom with a convex-analytic property: M-concave functions are exactly the restrictions to lattice points of concave functions on the base polytope whose linear perturbations are all maximized on integral base polytopes. It is the reason the M-concave class supports a convex-analysis-style theory at all: local optimality implies global optimality, maximizers of linear perturbations are well behaved, and conjugacy (the paper's Theorems 5.3 and 6.4, separate missions of this series) can be developed. Theorem 3.1 on its own is widely used to verify M-concavity in applications, since it reduces the exchange axiom to pairs at distance four.

All results here were proved in 1996. To our knowledge none of them has a machine-checked proof; Mathlib has matroids on sets but no integral base sets in $\mathbb Z^V$, no M-concave functions and no concave closure of a function on a finite set. A formal proof of this chain would be a first formal development of discrete convex analysis.

Difficulty

The equivalence of (EXC) with its local version (Theorem 3.1) is not a routine induction on $\|x-y\|$: the exchange inequality for a far pair does not follow from the inequalities along a path of distance-4 pairs, because the exchange partner $v$ must be chosen consistently with the prescribed $u$. For the "if" direction of Theorem 4.4, knowing that every maximizer set is an integral base set says nothing directly about the values of $\omega$ at non-maximizing points; turning this global information on maximizers into the local inequality (EXC$_{\mathrm{loc}}$) requires a supporting hyperplane of the concave closure at a well-chosen point and the integrality of the intersection of an integral base polytope with a box (a cited result on submodular systems). Theorem 4.6 then needs the concave closure to agree with $\omega$ on $B$ (Lemma 4.5), which fails for general $\omega$.

Formalization scope

All declarations live in the namespace SteinitzExchange.Extension. The ground set is a type V with [Fintype V] [DecidableEq V] [Nonempty V]; integer vectors are V → ℤ, real vectors V → ℝ, and toReal embeds the former into the latter. $B$ is a Finset (V → ℤ). A function $\omega:B\to\mathbb R$ is a total (V → ℤ) → ℝ whose values are only ever read at points required to be in $B$. $\overline B$ is Mathlib's convexHull ℝ of the image of $B$. Pinned readings:

1. Integral base polytope means the convex hull of a finite nonempty set satisfying (B1) (by the paper's Theorem 2.1 this is its meaning), not "a polytope with integer vertices".
2. The concave closure is a real infimum; it is the paper's value on $\overline B$ and a junk value $0$ off $\overline B$ (the paper's $-\infty$), so every statement uses it only on $\overline B$. $\operatorname{argmax}(\hat g)$ and $\operatorname{argmax}(\bar\omega[p])$ range over $\overline B$ only; the concave conjugate is a minimum over the nonempty finite $B$.
3. Lemma 3.2's maximum with $-\infty$ entries is stated as: for one of the two pairings both exchanged points lie in $B$ and the bound holds for that pairing.
4. Theorem 4.4 and Lemma 4.3 conclude that $\operatorname{argmax}(\omega[p])$ is itself an integral base set. Read literally ("its convex hull is an integral base polytope") the "if" direction of Theorem 4.4 is false: $B=\{(2,0),(1,1),(0,2)\}$ with $\omega=(0,-1,0)$ is a counterexample. The paper's proof, its gloss in Lemma 4.3 and its use on p. 292 all take the integral-base-set reading. Theorem 4.6 needs no such adjustment and is stated as printed.
5. Theorem 2.2 carries the standing assumption of §2.3 that $\omega$ satisfies (EXC).

Trivializing formalizations are ruled out: the extension $\bar\omega$ must agree with $\omega$ on $B$ and be concave on $\overline B$, the argmax is over $\overline B$ and not over $\mathbb R^V$, and an integral base polytope is never empty.

A complete development needs basic facts on integral base sets (the equivalence of (B1) with the simultaneous exchange (B2), $B=\mathbb Z^V\cap\overline B$, and the paper's cited Theorem 2.1 relating them to submodular functions), the representation (4.3) of the concave closure as a maximum of convex combinations, and supporting hyperplanes of polyhedral concave functions. The layer of integral base sets and M-concave functions is reusable for the two other missions of this series and for any later formalization of discrete convex analysis; contributions of general-purpose lemmas about it are welcome.

Selected references

• K. Murota, Convexity and Steinitz's exchange property, Advances in Mathematics 124 (1996) 272–311. https://doi.org/10.1006/aima.1996.0084
• K. Murota, Discrete Convex Analysis, SIAM Monographs on Discrete Mathematics and Applications, 2003. https://doi.org/10.1137/1.9780898718508

Call a set of reals a Sidon set when all its pairwise sums are distinct: if $a + b = c + d$ with all four in the set, then $\{a,b\} = \{c,d\}$.

The goal. There is an absolute constant $c > 0$ such that every finite set $X$ of positive reals contains a Sidon subset $S$ with $|S| \ge c\sqrt{|X|}$.

This is the lower bound half of Erdos problem 530, which Riddell posed and which asks for the order of the largest guaranteed Sidon subset. That problem is open: it asks whether the guarantee is asymptotically $N^{1/2}$, and the constant is not known. What is settled is the order, by Komlos, Sulyok, and Szemeredi, Linear problems in combinatorial number theory, Acta Math. Acad. Sci. Hungar. 26 (1975) 113-121, as a case of a general theorem about linear equations. Erdos had previously observed the cube-root lower bound and the matching $(1+o(1))N^{1/2}$ upper bound from $A = \{1, \dots, N\}$. A second and much shorter proof is in Bailleul and Riblet, arXiv:2605.03181.

The exponent is the whole problem

A one-paragraph argument gives $|S| \ge c|X|^{1/3}$: take a Sidon subset $S$ of maximum size, and note that every $x$ outside it satisfies $x = c + d - b$ or $x = (c+d)/2$ for elements of $S$, so $|X| \le 3|S|^3$.

That cube root is not a weak first attempt, it is the ceiling for any argument that only counts. An arithmetic progression of length $n$ has additive energy of order $n^3$, so a probabilistic argument cannot see the difference between it and a generic set. Getting from $1/3$ to $1/2$ requires using the structure of the set, and that is what both published proofs do.

The idea both proofs share

Compress, then pigeonhole against a known Sidon set.

An arbitrary finite set of reals has no arithmetic to work with, so first move it into $\mathbb{Z}$: a finite set spans a finite dimensional $\mathbb{Q}$-vector space, and a generic rational functional separates its points while preserving every relation $a + b = c + d$. Then squeeze the resulting integers into an interval of length comparable to their number, keeping a constant fraction of them and keeping the property that a Sidon subset of the image lifts to one of the original. Finally intersect with a translate of the Erdos-Turan Sidon set, which has about $\sqrt{N}$ elements inside $\{0, \dots, N-1\}$. A set of size $\Theta(n)$ inside $[1,n]$ meets some translate of a Sidon set of size $\sqrt{n}$ in order $\sqrt{n}$ points, and that intersection is Sidon.

The two proofs differ only in the compression step, and the mission carries both.

The two routes

The 1975 route compresses in four lemmas driven by a remainder map: choose a modulus $q$ dividing no difference, dilate so that the remainders are small, and observe that a small remainder map preserves $a + b = c + d$. Finding the modulus needs a prime counting bound.

The 2026 route replaces all four with one averaging lemma over a real rotation parameter $\theta$, keeping the elements whose fractional part of $am\theta$ is below $1/2$, where no carry occurs. No prime counting appears anywhere.

Notes on the formalization

Every item is stated in Mathlib primitives alone, so the mission needs no definition items. The Sidon condition, the Erdos-Turan construction, and the reduction relation of the 1975 route are all written out at each use.

The published 2026 proof finishes with Singer's 1938 covering of $\mathbb{Z}/(q^2+q+1)\mathbb{Z}$ by $q+1$ Sidon sets. Mathlib has no perfect difference sets, so the mission uses averaging over translates instead. It does the same job at the same order and gives a worse constant, which costs nothing because the goal asserts only that some $c > 0$ exists.

Two lemmas of the 1975 paper are deliberately absent. A local formalization of Lemma 2 and Lemma 6 turned out to be false as stated, machine-checked in both cases, so neither is offered here as a milestone. Those are errors in that rendering rather than in the paper, and the 2026 route reaches the goal without either. Lemma 1' is absent for the same practical reason: the 2026 route does not need it.

Motivation

A cooperative game with transferable utility assigns to every coalition of players the total payoff the coalition can secure on its own. Two solution concepts for such games go back to the foundations of game theory: the core, the set of payoff divisions no coalition can improve upon, and the stable set (von Neumann–Morgenstern solution), a set of divisions that is internally consistent and externally absorbing under the relation of domination. For general games the two concepts behave badly: the core may be empty, stable sets may fail to exist (Lucas 1968), and when they exist there are usually many of them.

Lloyd Shapley's paper Cores of Convex Games (Int. J. Game Theory 1, 1971) isolates a class of games, the convex games (supermodular characteristic functions), on which all of this becomes well behaved. Convex games arise in cost allocation, in bankruptcy and airport problems, in scheduling and sequencing games, and in any setting with increasing returns to cooperation; the supermodular functions behind them are the same objects studied as polymatroid rank functions in combinatorial optimization (Edmonds 1970). For such games the paper shows that the core is nonempty, that its faces fit together in a rigid combinatorial pattern, that its vertices are exactly the marginal-contribution vectors, and that the core is the unique stable set.

Setting

Let $N=\{1,\dots,n\}$ be a finite set of players. A game is a function $v$ from subsets of $N$ to the reals with $v(\emptyset)=0$. It is convex if

A payoff vector is $a\in\mathbb R^N$, and $a(S)=\sum_{i\in S}a_i$. It is feasible if $a(N)\le v(N)$. The core $C$ is the set of feasible $a$ with $a(S)\ge v(S)$ for every $S\subseteq N$; in particular $a(N)=v(N)$ on $C$.

For a nonempty coalition $S$, the face $C_S$ is the set of core points with $a(S)=v(S)$; by convention $C_\emptyset=C$, and $C_N=C$. The family $\{C_S\}$ is the core configuration. It is complete if no $C_S$ is empty, and regular if $C_N\ne\emptyset$ and

For an ordering $\omega$ of the players, $S_{\omega,k}$ is the set of the first $k$ players, and the marginal vector $a^\omega$ pays each player $i$ its marginal contribution $v(S_{\omega,\omega(i)})-v(S_{\omega,\omega(i)-1})$.

A payoff vector $b$ is dominated by $a$ if some nonempty coalition $S$ has $a(S)\le v(S)$ and $a_i>b_i$ for all $i\in S$. A set $V$ of feasible vectors is stable if every feasible vector is either a member of $V$ or dominated by a member of $V$, but not both.

Formalization targets

Goal: Theorem 8

The goal contains both halves of the page's statement: stability of the core, and uniqueness ("the unique von Neumann–Morgenstern solution").

Milestones, in the order the argument uses them

• Lemma 1 (p. 18) and Lemma 2 (p. 19): for a regular configuration, a point on two nested faces $C_S\cap C_T$ with $|T\setminus S|\ge2$ can be moved to a face $C_Q$ of an intermediate coalition, and a point of $C_S$ to $C_S\cap C_{S\cup\{j\}}$, keeping its coordinates on $S$.
• Theorem 2 (p. 18): in a regular configuration $C_{S_1}\cap\cdots\cap C_{S_m}\ne\emptyset$ for every strictly increasing chain $S_1\subset\cdots\subset S_m$; in particular a regular configuration is complete.
• Theorem 4 (p. 21): the core of a convex game is nonempty.
• Theorem 5 (p. 22): a game is convex if and only if its core configuration is regular.
• Two claims of §4.3 (p. 24): every stable set contains the core, and no stable set properly includes another.
• The claim that opens the proof of Theorem 8 (p. 24): in a convex game every feasible vector outside the core is dominated by a core point.

The mission also states Theorem 3 (p. 19), the vertices of a regular core are exactly the marginal vectors $a^\omega$, as a further item that is not on the path to the goal.

Significance

The result. Theorem 8 gives, for a natural and widely occurring class of games, a complete answer to the existence and uniqueness questions for von Neumann–Morgenstern solutions, which are open or negative in general. Theorems 3 and 5 describe the core of a convex game explicitly as the polytope spanned by the $n!$ marginal vectors, the combinatorial description that underlies later work on the Shapley value, the Weber set, and the polymatroid greedy algorithm. Theorem 5 is the geometric characterization of supermodularity through the face structure of the core.

Formalizing it. All results in this mission are proved in the paper; none has a machine-checked proof on the platform. Theorem 4 is already stated on the platform (as part of a statement that also puts every marginal vector and the Shapley value in the core) and enters the mission as an existing item. The remaining work is a formal development of face configurations of the core, of stable sets and domination, and of the passage from supermodularity to the geometry of the core. The definitions of stable set and domination are general and reusable for any transferable-utility game.

Difficulty

The internal half of stability is immediate from the definitions: a core point cannot be dominated by any vector satisfying a coalition constraint $a(S)\le v(S)$. Uniqueness also follows from two short observations. The substance is external stability: every feasible vector outside the core must be dominated by a core point, and the dominating vector has to be produced explicitly. The obvious attempt, raising the payoffs of one violated coalition and leaving the other coordinates of $b$ unchanged, does not in general produce a core point, and nothing in the definition of the core alone controls how the core meets the hyperplane of a given coalition; that control is what the face theory of §3 is about. For non-convex games the external half genuinely fails, so no argument that ignores convexity can succeed.

Formalization scope

Players are Fin n (a relabelling of the paper's finite set $N$), a game is f : Finset (Fin n) → ℝ, payoff vectors are Fin n → ℝ, and $a(S)$ is ∑ i ∈ S, a i. The existing platform definitions Supermodularity.Cooperative.IsConvexGame ($v(\emptyset)=0$ plus supermodularity on all subsets), Core, InitialCoalition and GreedyPayoff (the marginal vectors, orderings being permutations of Fin n) are reused; the reused Theorem 4 statement is Supermodularity.Cooperative.convex_game_core_and_shapley.

Conventions committed to:

• Wherever the page says "a game", the hypothesis is exactly $v(\emptyset)=0$; convexity is IsConvexGame.
• Faces satisfy $C_\emptyset=C$ literally: the tightness condition is imposed only for nonempty $S$.
• Regularity includes $C_N\ne\emptyset$, as on the page.
• Lemmas 1–2 and Theorems 2–3 assume a regular configuration, not convexity, as on the page.
• $S\subset\subset T$ is $S\subsetneq T$ with $|T|-|S|\ge2$; Lemma 1's two preassigned elements are distinct.
• An increasing sequence of $m\ge1$ coalitions is a strictly monotone map from Fin (m + 1).
• "Vertex" is Set.extremePoints ℝ.
• Domination requires a nonempty coalition and strict coordinate inequalities; stable sets consist of feasible vectors and the "either … or …, but not both" condition ranges over feasible vectors, following the page rather than the classical imputation-based variant.

A formalization in which the dominating coalition may be empty, in which regularity omits $C_N\ne\emptyset$, or in which the goal asserts stability without uniqueness does not state the paper's theorem and is ruled out.

Welcome contributions: proofs of the milestones in any order, general lemmas about faces of polytopes cut out by set-function inequalities, and reusable API for domination and stable sets.

Selected references

• L. S. Shapley, Cores of Convex Games, International Journal of Game Theory 1 (1971), 11–26. https://doi.org/10.1007/BF01753431
• J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944.
• J. Edmonds, Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and Their Applications, Gordon and Breach, 1970, 69–87. https://doi.org/10.1007/3-540-36478-1_2
• W. F. Lucas, A game with no solution, Bulletin of the American Mathematical Society 74 (1968), 237–239. https://doi.org/10.1090/S0002-9904-1968-12039-2
• D. M. Topkis, Supermodularity and Complementarity, Princeton University Press, 1998, §5.2.

Call $A \subseteq \mathbb{Z}/2^n\mathbb{Z}$ cube-free if no triple $x, y, z$ has all seven of $x$, $y$, $z$, $x+y$, $y+z$, $z+x$, $x+y+z$ inside $A$. The triple is unconstrained, so a degenerate one counts. Write $f(n)$ for the largest size of a cube-free subset.

The conjecture. $f(n) \le \frac{5}{8} 2^n$ for every $n$.

This is Conjecture 5.1 of Jason Long and Adam Zsolt Wagner, The largest projective cube-free subsets of $\mathbb{Z}_{2^n}$, arXiv:1810.01225. It has been open since October 2018, and a 2026 journal paper still names it as conjectured: Yuchen Meng, On Cube-Free Problems, Electron. J. Combin. 33(1) (2026) #P1.16.

The constant is attained

The bound is sharp, and the extremal set is explicit: $A = \{v : v \bmod 8 \in \{1,3,4,5,7\}\}$, the odd residues together with those congruent to 4 mod 8. Its size is $2^{n-1} + 2^{n-3} = \frac{5}{8} 2^n$. In the layer language of Long and Wagner this is $C_3 = L_1 \cup L_3$.

What is known

The conjecture holds for unions of layers. That is Long-Wagner Theorem 1.10 at $d = 3$, and it is the largest class on which the conjectured constant is proved.

For arbitrary sets the best published unconditional bound is $f(n) < \frac{2}{3} 2^n$. Meng calls this bound "quite trivial" and gives it in one paragraph for every cyclic group, so it should not be read as progress toward $5/8$. The residual gap is exactly $\frac{2}{3} - \frac{5}{8} = \frac{1}{24}$, that is $2^n/24$ elements.

Small values are $f(1) = 1$, $f(2) = 2$, $f(3) = 5$, $f(4) = 10$, $f(5) = 20$, $f(6) = 40$, $f(7) = 80$, matching $2^{n-1} + 2^{n-3}$ from $n = 3$ on.

State those values honestly. They come from solver searches, Gurobi in Long and Wagner for $n \le 7$ and an independent SAT reproduction. The SAT half that matters, the unsatisfiability of "a cube-free set of size 81 exists at $n = 7$", is a solver claim with no proof certificate checked and no kernel check behind it. The witness half is verified: a set of exactly 80 elements was produced and re-checked cube-free. So $f(7) \ge 80$ is solid and $f(7) \le 80$ is not certified. Nothing in this mission rests on either.

What the items are

The goal item is the conjecture itself, for $n \ge 4$, and it is open. Every other item is a milestone that is proved mathematics, and the two closed instances $n = 4$ and $n = 5$ are stated separately because they are the only cases of the goal that a proof assistant has actually settled here.

The chain runs: the base case mod 8 by exhaustion, monotonicity under subsets, the bridge between the membership form and the Finset form of the forbidden configuration, sharpness, the odd-residue tight case, the layer-union theorem, the two-thirds bound, and then $n = 4$ and $n = 5$.

Notes on the formalization

Six definitions live in one definition item, Def_Z2nCubeFreeLayers: HasCube, CubeFree, config, ConfigFree, layerIdx and IsLayerUnion. layerIdx is written through the 2-adic valuation rather than through a congruence, because the congruence form leaves $0$ in no layer at all and needs the last layer special-cased.

CubeFree and ConfigFree are two encodings of the same condition and they are not definitionally equal, because config collapses duplicates on a degenerate triple. Their equivalence is a milestone rather than an assumption.

Motivation

A matching cut is a vertex bipartition whose crossing edges form a matching. The property was introduced under the name decomposability and has links to graph algorithms, stable cutsets in line graphs, and several graph-labeling problems. An Open Problem Garden question asked whether sufficiently large girth forces a matching cut once average degree is bounded.

Feghali, Lucke, Paulusma, and Ries answered that question negatively. Their conference paper appeared at ISAAC 2023, and the version of record was published in Algorithmica in 2025. The paper proves NP-completeness for bipartite graphs of arbitrarily prescribed girth and bounded maximum degree. A central input, Lemma 5, is a stronger structural existence statement: for every girth threshold there is an immune $14$-regular bipartite graph of at least that girth, and it has a perfect matching.

This is therefore a ResearchPaper mission, not a new open-problem mission. Its goal is to formalize the published theorem and its graph-theoretic consequence. Repository candidate constructions and finite arithmetic audits remain separate and are not credited as solving the problem.

Setting

For a finite simple graph $G$ and a vertex set $A\subseteq V(G)$, the associated cut consists of all edges with one endpoint in $A$ and one in $V(G)\setminus A$. The cut is nontrivial when both shores are nonempty. It is a matching cut when each vertex is incident with at most one crossing edge. A graph is called immune in the cited paper when it has no matching cut.

The girth is the length of a shortest simple cycle; forests have infinite girth. A graph is $14$-regular when every vertex has exactly fourteen neighbors. Bipartiteness is witnessed by a partition into two independent sides. A perfect matching pairs every vertex with one adjacent partner.

The original OPG wording has a literal one-vertex boundary ambiguity: with nonempty shores required, $K_1$ has no matching cut, average degree zero, and infinite girth. The research-paper target avoids that vacuity by constructing connected graphs with at least two vertices, exact degree fourteen, and arbitrarily large finite girth.

Formalization targets

Lemma 5 — immune high-girth graphs

The main theorem follows the paper's structural lemma:

The graph may depend on $g$. The existence quantifier does not request an efficient algorithm or a numerical order bound.

Negative OPG consequence

A supporting theorem removes the perfect-matching and bipartite fields and records the direct substantive counterexample family:

Thus choosing $d=15$ refutes the intended universal assertion that some girth threshold works for every graph of average degree below $d$.

Significance

The theorem shows that large girth and bounded degree do not force matching cuts. The examples are highly nontrivial: they are connected, regular, bipartite, and can have arbitrarily large girth. This separates local tree-like structure from the global expansion that prevents a matching cut.

Within the paper, the immune graphs serve as gadgets for hardness reductions. The journal theorem states that, for every $g\ge3$, Matching Cut is NP-complete even for bipartite graphs of girth at least $g$ and maximum degree at most $60$. Formalizing Lemma 5 supplies the graph-theoretic core needed to reconstruct that result without forcing this mission to formalize an entire complexity-theory reduction in its first stage.

Difficulty

Large girth alone makes bounded neighborhoods look like trees, and trees have many matching cuts. Immunity must therefore come from global expansion rather than short local cycles. The paper obtains the required family from Lubotzky–Phillips–Sarnak Ramanujan graphs and combines spectral and isoperimetric bounds to show that every nontrivial cut has too many crossing incidences to be a matching.

A formal proof must bridge several exact interfaces: existence of suitable primes, the finite Cayley-graph construction, bipartiteness and regularity, the girth lower bound, the spectral-to-isoperimetric inequality, and Hall's theorem for the perfect matching. None of these can be replaced by a finite sample or an asymptotic slogan.

Formalization scope

Graphs are finite and simple. Connectedness is nonempty mutual graph reachability. A simple cycle is a cyclic list of at least three distinct vertices; girth at least $g$ means every such cycle has length at least $g$, so forests satisfy every threshold. A matching cut requires both shores nonempty and is encoded by the condition that every vertex has at most one crossing neighbor. A perfect matching is represented by an adjacent involution.

The main theorem explicitly requires at least two vertices, although exact 14-regularity already forces nontrivial order; the redundant bound documents exclusion of the $K_1$ ambiguity. The mission does not claim that the frozen OPG contract was well-posed at order one. It formalizes the paper's substantive counterexample family and the consequence for the intended question.

Candidate files in the associated repository explore alternative bounded-degree constructions and integer counts. They are candidate_only and are not proof dependencies. Contributions should follow the published Lemma 5 and its cited inputs, or provide a separately sourced proof of the same declaration. The later maximum-degree-60 NP-completeness theorem is welcome as a future extension after the finite complexity framework is fixed.

Selected references

• C. Feghali, F. Lucke, D. Paulusma, and B. Ries, Matching Cuts in Graphs of High Girth and H-Free Graphs, Algorithmica 87 (2025), 1199–1221. https://doi.org/10.1007/s00453-025-01318-8
• C. Feghali, F. Lucke, D. Paulusma, and B. Ries, ISAAC 2023 version. https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.31
• A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261–277. https://doi.org/10.1007/BF02126799
• Open Problem Garden, Matching cut and girth. https://www.openproblemgarden.org/op/matching_cut_and_girth

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Status update (2026-09-08): The root theorem OPG500Counterexample.eight_vertex_counterexample is now Proved by an accepted Prove2Me submission. All six milestones are proved and the root has zero open leaves. The accepted proof has also been independently rebuilt with Lean 4.33.1 / Mathlib 0df444a360eaa60ab8c11dca51a86af692955474. The historical text below describes the mission as it stood before formal closure.

Motivation and historical context

Peripheral cycles occupy a distinguished place in structural graph theory. A cycle is peripheral when it is induced and does not separate the graph after its vertices are removed. Tutte proved in 1963 that the peripheral cycles of a finite 3-connected graph generate its binary cycle space. This theorem links a local, visibly embedded kind of cycle to the global algebraic structure of all cycles.

Weighted geodesic cycles provide a different generating family. Georgakopoulos and Sprüssel proved in 2009 that, for every finite graph with positive edge lengths, every cycle is a binary sum of weighted geodesic cycles whose lengths do not exceed the length of the original cycle. In the same paper they posed Problem 3: can the edges of every finite 3-connected graph be assigned positive lengths so that every weighted geodesic cycle is peripheral? A positive answer would recover Tutte's generation theorem through metric structure.

The present target tests the opposite possibility on one explicitly specified graph with eight vertices. A candidate argument and finite certificates are available in the frozen OPG-500 research repository, but those artifacts are explicitly marked candidate_only: they are neither a published counterexample nor a machine-checked resolution. The purpose of the formal target is to determine whether the proposed universal obstruction survives complete definition, proof, and statement-faithfulness checks.

Setting

Let $G$ be a finite simple graph. A positive edge-length assignment is a function

such that $\ell(e)>0$ for every edge $e$. The length of a finite path or cycle is the sum of the lengths of its edges.

A simple cycle $C$ is $\ell$-geodesic when, for every pair of vertices $x,y$ on $C$, at least one of the two $x$–$y$ arcs of $C$ has length equal to the shortest-path distance between $x$ and $y$ in $G$. Equivalently, there is no $x$–$y$ path in $G$ whose length is strictly smaller than both $x$–$y$ arcs of $C$. The definition concerns vertices of the cycle and permits ties between shortest paths.

A simple cycle is peripheral when it is induced and deleting all of its vertices leaves a connected graph or the empty graph. This is vertex deletion, not edge deletion.

Fix the graph $H$ on vertices $0,1,\ldots,7$. The vertices $0,1,2,3$ induce $K_4$. For each $i\in\{0,1,2,3\}$, set $y_i=7-i$ and join $y_i$ to exactly the three core vertices other than $i$. The four vertices $y_i$ are pairwise nonadjacent. Thus the frozen edge set is

The labels and edge set are part of the statement and are not interchangeable with earlier candidate labelings without an explicit isomorphism.

Formalization targets

Main target: the universal eight-vertex obstruction

Formalize the following statement for the fixed graph $H$:

The existential cycle may depend on $\ell$. The universal quantifier includes all strictly positive real assignments, including assignments with tied shortest paths. This is the stable target; finite samples and rational specializations are subordinate checks rather than replacements for it.

Supporting targets

The development should also formalize the finite weighted geodesic-cycle generation theorem of Georgakopoulos and Sprüssel, the exact 3-connectivity and peripheral-cycle classification of $H$, the required shortest-path and tight-subgraph statements, the finite cycle-space rank statements, and the finite minimum/descent principle used to select a cycle outside a closed binary span. These targets should remain separate declarations so that their assumptions and reuse boundaries are visible.

Significance

A proof of the main target would give a negative answer to the finite problem by exhibiting a 3-connected graph for which no positive edge weighting can make all geodesic cycles peripheral. It would not contradict Tutte's theorem: peripheral cycles may still generate the cycle space even though they cannot be made to contain every geodesic cycle for any weighting. The distinction between these two generation mechanisms is part of the mathematical content.

A formal development would add more than a checked final sentence. It would provide reusable definitions for positively weighted finite graphs and vertex-geodesic cycles, a precise treatment of the two arcs between cycle vertices, explicit deletion semantics for peripheral cycles, and finite cycle-space infrastructure. It would also separate purely finite graph facts from statements quantified over arbitrary real weights. The candidate repository currently supplies finite enumeration and abstract Lean fragments, but no existing artifact checks this full dependency chain.

Until the complete main theorem is verified, the eight-vertex graph remains a candidate obstruction and the original problem remains unresolved by this development.

Difficulty

The central difficulty is the universal quantification over real edge lengths. Testing many integer or rational vectors cannot cover it. Shortest paths need not be unique, so an argument that silently perturbs the weights or assumes unique geodesics can change which cycles are geodesic. Every strict and weak inequality must therefore agree with the source definition, including tie cases.

The graph is small but the semantic boundary is not. A formal cycle representation must expose the two cycle arcs for every vertex pair without admitting malformed or repeated-vertex objects. The peripheral predicate must combine inducedness with connectivity after vertex deletion and must classify all cycles, not only a selected family of triangles. Finally, finite cycle-space computations and rank inequalities must be connected to actual paths and weighted geodesicity; a propositional or enumerative certificate alone does not establish that bridge.

Formalization scope

The Lean development will use Fin 8 for the vertices of $H$ and a SimpleGraph representation for adjacency. Weights will be functions on the edge subtype, so values on nonedges cannot affect the theorem. All weights are real and strictly positive. Paths and cycles are finite and simple; arbitrary walks do not count as target witnesses. Geodesicity is vertex-based and includes tied shortest paths. Peripheral cycles use inducedness and vertex deletion, with a connected-or-empty remainder.

The main theorem must retain the quantifier order “for every weighting, there exists a cycle.” It may not be weakened to rational weights, finitely many tested assignments, nonnegative weights, one selected weighting, edge-geodesicity, or the assertion that only four named core triangles fail to be peripheral. Definitions must be sorry-free, and nontrivial mathematical claims must be theorem declarations with separately checked proofs.

Reusable contributions include finite weighted-path length, shortest-path attainment in finite positive graphs, the equivalence of the two geodesic formulations, cycle-arc APIs, vertex-deletion connectivity, binary edge-vector encodings, and finite descent outside a closed span. Graph-specific finite certificates are welcome only when their checker is represented in Lean or their conclusions are otherwise proved in the kernel.

Selected references

• A. Georgakopoulos and P. Sprüssel, Geodetic topological cycles in locally finite graphs, Electronic Journal of Combinatorics 16 (2009), R144. Section 3.1, Theorem 3.1; Section 5, Problem 3. https://arxiv.org/abs/0911.3999v1
• Open Problem Garden, Geodesic cycles and Tutte's Theorem, problem statement and vertex-based definition. https://www.openproblemgarden.org/op/geodesic_cycles_and_tuttes_theorem
• W. T. Tutte, How to draw a graph, Proceedings of the London Mathematical Society 13 (1963), 743–768. Cited as reference [18] by Georgakopoulos and Sprüssel for peripheral-cycle generation.
• Vibe Mathing, frozen OPG-500 candidate repository at commit a41fe59b4535851ea55f6e868e938b9aaf81e924. https://github.com/vibemathing/problem-opg-500-geodesic-cycles/tree/a41fe59b4535851ea55f6e868e938b9aaf81e924

Closed — negative resolution of Erdős Problems 96 and 97

Adam McKenna closed this mission on 13 September 2026 following Unit distances in convex polygons, by Liam Kruer, Jensen Kohlmeyer, and Liam Price. Their construction gives strictly convex point sets with Ω(n log log n) unit-distance pairs and arbitrarily large minimum unit-distance degree, answering both questions and the general fixed-k version of Problem 97 negatively.

Paper and complete Lean source. All credit for the counterexample and its formalization belongs to those authors. Adam McKenna prepared the Prove2Me adapters.

Do not start further proof attempts or solver runs for the affirmative conjectures. Existing statements, conditional lemmas, partial proofs, and milestones remain as historical work. The owner has authorized closure assuming the external result is correct; individual theorem pages report Prove2Me verification status.

Historical mission description

Motivation

The mission is to prove the combined open goal

for finite point sets in strictly convex position in the Euclidean plane.

Why Problems 97 and 96 belong together

Problem 97 gives the local step needed for Problem 96. Assume Problem 97. Every nonempty convex-independent finite set then has a vertex with at most three neighbors at each positive radius, in particular at radius $1$. Delete that vertex and preserve convex independence. Apply the same step to every subset created by deletion until no points remain. Charge each unordered unit-distance pair to the first endpoint deleted. Each deleted vertex receives at most three charges, so an $n$-point set determines at most $3n$ unordered unit-distance pairs. This gives the Problem 96 bound and therefore $O(n)$. The package uses this one-way dependency; it does not seek a reverse implication.

Setting

Let $A\subset\mathbb R^2$ be finite. Strict convex position means that every point of $A$ is an extreme point of the convex hull of $A$. For $p\in A$, the pinned multiplicity at radius $r>0$ counts points $q\in A$ with $\lVert p-q\rVert=r$. Problem 97 asks for a point where no radius has four such other points. Problem 96 counts unordered pairs at distance $1$, then takes the supremum over convex-independent $n$-point sets.

The historical progression is part of the setting. Erdős’s 1946 paper posed an earlier three-neighbor version. His 1987 account reports Danzer’s convex nonagon in which every vertex has three equidistant witnesses, and asks about four witnesses. Fishburn and Reeds’s 1992 work gives a 20-vertex convex configuration with the same unit distance at every vertex, placing the local question beside the unit-distance problem.

Target

The Problem 97 target is the canonical statement that every nonempty finite convex-independent $A$ has no four-equidistant-point property:

The Problem 96 target is the canonical asymptotic statement

where $U_c(n)$ is the supremum of the unordered unit-distance counts determined by convex-independent $n$-point sets. The bound is asymptotic; the Problem 97 route would give the stronger explicit bound $U_c(n)\le3n$ for every natural number $n$.

Significance

The package records a formal proof route joining a pinned geometric obstruction to a global extremal bound. A successful Problem 97 proof would immediately settle Problem 96 with the explicit constant $3$, while preserving the combinatorial meaning of the count. It also separates the historical three-neighbor constructions from the still-open four-neighbor assertion.

Difficulty

The source proof reduces Problem 97 to strong induction on $|A|$. Its counting engine follows Dumitrescu's 2006 isosceles-count method, with the cap-witness refinements used in the source attributed to Nivasch--Pach--Pinchasi--Zerbib (2013). This engine forces every counterexample to have at least nine points; a finite geometric analysis excludes exactly nine points; and the remaining step must produce a removable vertex for every larger minimal counterexample. The removable-vertex statement carries the induction hypothesis that every strictly smaller nonempty convex 4-equidistant set is contradictory. That large-cardinality geometric step remains open, so both headline targets remain open. Finite computational certificates can support local cases but do not replace the universal geometric statement.

Counterexample routes

Problem 97 is open, so the mission also records the parallel negative route. The source formalization calls a nonempty convex-independent finite set with the four-equidistant property a Problem97.IsCounterexample. Constructing one such set would refute Problem 97 and therefore refute the mission's affirmative conjunction, regardless of whether Problem 96 remains true. The counterexample milestone keeps this resolution path visible beside the nonexistence proof. A successful witness must use exact coordinates or exact algebraic data from which Lean verifies both strict convex position and the four-equidistant property; a numerical approximation or a realizable incidence pattern alone is insufficient.

Problem 96 has its own negative route. Because its claim is asymptotic, one finite convex configuration cannot refute it. A counterexample must instead give convex-independent point sets at arbitrarily large cardinalities whose unit-distance counts exceed every proposed linear constant. The mission tracks this superlinear-family statement separately, together with a reduction from it to the exact negation of Problem 96. This keeps both possible outcomes visible: a direct or Problem-97-derived linear upper bound, and an explicit family proving that no such bound exists.

Formalization scope

The canonical source is pinned at commit 757d852766f377f7c1a0ffeeef6d3526bc0cb7a4. It contains the formal source statements for Problem 97 and Problem 96. The source repository reports closed proofs of the conditional bridge to the $3n$ bound (conditional three-times bound), the $|A|\ge9$ counting milestone (nine-point counting bound), and the exact nine-point exclusion (exact nine-point exclusion theorem). The remaining large-cardinality milestone is the removable-vertex step, with its minimality hypothesis retained. The current platform mission contains accepted transfers of the counting argument, the conditional bridge, and the exact nine-point exclusion, while the removable-vertex step remains open. Its definitions make convex independence and the positive-radius condition explicit; no theorem is assumed inside a definition. Singletons and two-point sets are included in Problem 97, while Problem 96's counting definitions also include the empty set. The source repository uses Lean v4.27.0; these mission statements target the platform's v4.33.1. Source-proof transfer and revalidation remain separate work. The Lean declarations and proofs are this project's own formalization. The Dumitrescu and Nivasch--Pach--Pinchasi--Zerbib citations record mathematical provenance; they do not indicate that a paper proof was imported or machine-checked directly.

These source results establish the intended dependency graph: the P97 universal root feeds low-unit-degree extraction, strong induction, and then the P96 supremum bound. The platform mission records those contracts and milestones; it does not claim to have transplanted their proof bodies. The milestones include the two canonical roots, their conditional bridge, the |A| ≥ 9 count, the n = 9 exclusion, the |A| > 9 removable-vertex step, the documented Danzer nine-point three-neighbor example, the parallel goal of constructing a Problem 97 counterexample, and the superlinear-family route to a counterexample to Problem 96.

References

• Erdős, On Sets of Distances of n Points (1946), DOI.
• Erdős, Some Combinatorial and Metric Problems in Geometry (1987), scan.
• Fishburn–Reeds, Unit Distances Between Vertices of a Convex Polygon (1992), publisher record.
• Dumitrescu, On Distinct Distances from a Vertex of a Convex Polygon (2006), Springer record; provenance for the source counting method.
• Nivasch–Pach–Pinchasi–Zerbib, The Number of Distinct Distances from a Vertex of a Convex Polygon (2013), arXiv:1207.1266; provenance for the cap-witness refinements used by the source formalization.

Determine the exact second-order term in the least possible largest factor in a factorization of $n!$ into distinct integers exceeding $n$, with the proposed rational constant $4029639598/25970038185$.

Motivation

Graphs are the most basic combinatorial model of pairwise relations: road networks, frequency interference between radio stations, schedules, circuit layouts. Chapter 5 of Keller and Trotter's Applied Combinatorics (appliedcombinatorics.org, CC BY-SA 4.0) introduces the vocabulary of graph theory and proves its first structural theorems: when a graph can be traversed edge by edge (Euler, 1736), when it can be toured vertex by vertex (Dirac, 1952), when it can be colored with two colors, and how far the chromatic number can drift from the size of the largest clique.

Hamiltonicity is the vertex analogue of Euler's edge-traversal problem and behaves very differently: Euler's problem has a simple parity characterization, while deciding whether a graph has a hamiltonian cycle is NP-complete (Karp, 1972). Sufficient conditions are therefore the main tool, and the minimum-degree condition of Dirac (1952) is the first and most cited of them; Ore's condition (1960) and the Bondy–Chvátal closure (1976) refine it.

Setting

A graph $G = (V, E)$ consists of a finite vertex set $V$ and a set $E$ of 2-element subsets of $V$, the edges; $xy \in E$ means $x$ and $y$ are adjacent. The degree $\deg_G(v)$ is the number of neighbours of $v$. The mission uses Mathlib's SimpleGraph V with [Fintype V]; "a graph on $n$ vertices" means Fintype.card V = n.

• A cycle is a sequence $(x_1, \dots, x_n)$ of $n \ge 3$ distinct vertices with $x_i x_{i+1} \in E$ for $i < n$ and $x_1 x_n \in E$; its length is $n$. A graph is acyclic if it has no cycle, and a tree if it is connected and acyclic. A leaf of a tree is a vertex of degree $1$.
• A hamiltonian cycle is a sequence $(x_1, \dots, x_n)$ in which every vertex appears exactly once, $x_i x_{i+1} \in E$ for $i < n$, and $x_1 x_n \in E$. A graph is hamiltonian if it has one (AppliedComb.Graphs.IsHamiltonian).
• An eulerian circuit is a sequence $(x_0, \dots, x_t)$, repetition allowed, with $x_0 = x_t$, consecutive entries adjacent, and every edge equal to $x_i x_{i+1}$ for exactly one $i < t$. A graph without isolated vertices is eulerian if it has one (IsEulerian).
• A proper coloring assigns colors to vertices so that adjacent vertices differ; the chromatic number $\chi(G)$ is the least number of colors in a proper coloring, and the clique number $\omega(G)$ is the largest size of a set of pairwise adjacent vertices.
• An interval graph is the intersection graph of closed intervals $[a_v, b_v] \subset \mathbb R$, $v \in V$: distinct $u, v$ are adjacent iff their intervals meet (IsIntervalGraph).

Formalization targets

Goal: Dirac's theorem (Theorem 5.18)

Milestones

In the book's order:

1. Proposition 5.11. A tree on $n \ge 2$ vertices has at least two leaves.
2. Theorem 5.13 (Euler). A graph without isolated vertices is eulerian if and only if it is connected and every degree is even.
3. Theorem 5.21. $\chi(G) \le 2$ if and only if $G$ contains no odd cycle.
4. Proposition 5.24 (generalized pigeonhole). If $f : X \to Y$ and $|X| \ge (m-1)|Y| + 1$, some fibre of $f$ contains $m$ distinct elements.
5. Proposition 5.25. For every $t \ge 3$ there is a finite graph $G_t$ with $\chi(G_t) = t$ and $\omega(G_t) = 2$.
6. Theorem 5.28. Every finite interval graph satisfies $\chi(G) = \omega(G)$.

The book's proof of Theorem 5.18 uses only the pigeonhole principle, so none of these results lies on its path. The milestones are the chapter's other theorems about the same objects (cycles, degrees, colorings), and they share the goal's definitions. Cayley's formula (Theorem 5.39, $n^{n-2}$ labelled trees on $n$ vertices) is already on the platform as GYGraphTheory.cayley_tree_formula and is included as a reference item.

Significance

Dirac's theorem is sharp: the complete bipartite graph $K_{k,k+1}$ has minimum degree $k = \lceil n/2 \rceil - 1$ and no hamiltonian cycle. It is the starting point for the theory of degree conditions for hamiltonicity (Ore, Pósa, Chvátal) and for its extremal and random-graph analogues. Euler's theorem gives linear-time recognition of eulerian graphs; Theorem 5.21 characterizes bipartite graphs; Proposition 5.25 shows that local sparsity (no triangles) does not bound the chromatic number; Theorem 5.28 is the first step towards perfect graphs.

All of these results have been proved for a long time. What is missing is their formalization. Mathlib provides SimpleGraph, chromaticNumber, cliqueNum, degree-sum identities, and definitions of eulerian and hamiltonian walks. It contains no Dirac theorem, no converse of the eulerian degree condition, and no interval-graph theory. On the platform, FamousTheorems.two_colorable_iff_no_odd_cycle_6b is stated with odd closed walks rather than odd cycles, and triangle_free_chromatic_number asserts only $\chi \ge k$ rather than $\chi = t$ with $\omega = 2$ exactly.

Difficulty

For Dirac's theorem the obvious approach, induction on $n$ (deleting a vertex), fails: deleting a vertex lowers degrees, and the hypothesis $\deg \ge \lceil n/2 \rceil$ is not inherited by the smaller graph. The degree condition is global, and so is the argument that uses it. In Lean the argument also has to reverse and splice vertex sequences while keeping distinctness and every adjacency along the way. The sufficiency half of Euler's theorem has to construct a circuit that uses every edge exactly once, not just show that one exists up to parity. In Theorem 5.21 the obstruction is a cycle with distinct vertices; producing one from a failed 2-coloring takes more work than producing an odd closed walk. Proposition 5.25 needs a concrete infinite family of graphs and exact computation of both invariants. The upper bound $\chi \le t$ is easy, but the lower bound $\chi \ge t$ is not.

Formalization scope

• Graphs are SimpleGraph V on a Fintype V. The number of vertices is always Fintype.card V, never a free parameter; in Theorem 5.18 the ceiling $\lceil n/2 \rceil$ is (n + 1) / 2 in $\mathbb N$, with Fintype.card V = n.
• "Hamiltonian" is the book's definition (p. 79), a list of all vertices without repetition with consecutive and first/last entries adjacent. It is not Mathlib's Walk.IsHamiltonianCycle, which needs at least three vertices: under that notion Theorem 5.18 would be false for $K_2$. The goal assumes $V$ nonempty, as the book's proof does ($n$ positive). Degenerate readings are ruled out: a predicate satisfied by a cycle through only some vertices, or a statement in which $n$ is not the number of vertices, would make the goal vacuous or false, and neither is used here.
• "Eulerian" follows p. 75. Since the book defines it only for graphs without isolated vertices, Theorem 5.13 carries that hypothesis explicitly.
• Cycles are the book's cycles (distinct vertices, length $\ge 3$), not closed walks. Theorem 5.21 is stated with odd cycles.
• $\chi$ is Mathlib's chromaticNumber ($\mathbb N_\infty$-valued) and $\omega$ is cliqueNum; on finite graphs both agree with the book's definitions (pp. 81, 84).
• Planarity (Section 5.5: Euler's formula 5.32, the bound $3n - 6$ in 5.33, Kuratowski 5.34, the Four Color Theorem 5.37) is excluded. The book defines planar drawings through polygonal arcs in $\mathbb R^2$ and faces, which is a topological development rather than a chapter mission. Kuratowski's theorem and the Four Color Theorem are not proved in the book.
• Contributions reusable beyond this mission are welcome: path/cycle manipulation lemmas for list-based sequences, the Euler circuit construction, bipartiteness via distance parity, and the Mycielski construction.

Selected references

• M. T. Keller and W. T. Trotter, Applied Combinatorics, 2017 Edition, Chapter 5. https://www.appliedcombinatorics.org/
• G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. (3) 2 (1952), 69–81. https://doi.org/10.1112/plms/s3-2.1.69
• L. Euler, Solutio problematis ad geometriam situs pertinentis, Commentarii Academiae Scientiarum Petropolitanae 8 (1741), 128–140 (read 1736). https://scholarlycommons.pacific.edu/euler-works/53/
• J. Mycielski, Sur le coloriage des graphs, Colloquium Mathematicum 3 (1955), 161–162. https://doi.org/10.4064/cm-3-2-161-162
• R. M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations (1972), 85–103. https://doi.org/10.1007/978-1-4684-2001-2_9

Motivation

Paging is the problem of managing a fast memory that holds $k$ pages out of $n$: when a requested page is not in fast memory (a page fault), some resident page must be evicted, and the cost of an algorithm is its number of faults. Practitioners have many eviction rules. Least-recently-used (LRU) performs well on real workloads but can be $k$ times worse than the optimal off-line schedule; the randomized marking algorithm of the same paper is $2H_k$-competitive and so has better worst-case guarantees. Fiat, Karp, Luby, McGeoch, Sleator and Young asked in 1991 whether one on-line algorithm can combine the advantages of several given ones, and answered the question exactly: the attainable combinations of ratios are characterized by one inequality (arXiv:cs/0205038, §6).

The question of combining on-line algorithms has since become a theme of its own: combining heuristics with worst-case-safe algorithms, and, more recently, combining machine-learned predictions with robust fallbacks, both ask for the same kind of guarantee against several reference algorithms at once.

Setting

A type $(k,n)$ consists of $k$ servers and a finite set $M$ of $n$ vertices with the uniform metric: two distinct vertices are at distance $1$. This is paging: vertices are pages, the vertices covered by servers are the pages in fast memory, and a server move is a page fault.

A deterministic on-line algorithm $A$ of type $(k,n)$ has an initial configuration of its $k$ servers and, after each request $r\in M$, moves servers so that some server covers $r$; its configuration after a request sequence depends only on that sequence. Its cost $C_A(\sigma)$ on a request sequence $\sigma$ is the total distance its servers travel, i.e. the number of server moves.

For algorithms $A$ and $B$ of the same type and a constant $c$, $A$ is $c$-competitive against $B$ if there is a constant $a$ such that for every request sequence $\sigma$