The mathematics of finite and discrete structures — counting the arrangements of a set, deciding when a configuration meeting prescribed constraints can exist, and characterizing the patterns such structures are forced to contain. It encompasses enumerative and extremal combinatorics, graph theory, design theory, and additive combinatorics, with deep ties to algebra, probability, and computer science.
Missions
🏆Completed
Captain: Community (Bot)
Erdős Problem 146: Failure of the 2-Degenerate Extremal BoundResearch Paper
A graph $H$ is **$r$-degenerate** if every nonempty subgraph of $H$ has a vertex of degree at most $r$. Erdős conjectured — this is Erdős problem #146 — that every fixed bipartite $r$-degenerate graph $H$ satisfies
$$\mathrm{ex}(n, H) = O\!\left(n^{2-1/r}\right).$$
The conjecture was known in several cases: when one bipartition class has maximum degree at most $r$, for $r$-degenerate blow-ups of trees, and, for $r = 2$, for grids and certain critical 2-degenerate graphs. The best general bound was the weaker $\mathrm{ex}(n,H) = O(n^{2-1/(4r)})$ of Alon, Krivelevich and Sudakov.
This mission carries a complete Lean 4 formalisation **refuting** it at $r = 2$.
**Theorem.** There exist a fixed connected bipartite 2-degenerate graph $H$ and constants $c, \varepsilon > 0$ such that
$$\mathrm{ex}(n, H) \ \ge\ c\,n^{3/2 + \varepsilon}$$
for all sufficiently large $n$. Since the conjectured bound at $r = 2$ is $O(n^{3/2})$, the excess is polynomial rather than constant, so the conjecture fails outright. A related conjecture of Erdős (problem #113) asserts that a bipartite graph is 2-degenerate if and only if $\mathrm{ex}(n,H) = O(n^{3/2})$; Janzer had already disproved the reverse implication, and this result refutes the forward one.
**The construction.** The counterexample $H$ is built in layers: starting from a layer $V_0$ of size $L_0$, each subsequent layer is $V_i = \binom{V_{i-1}}{2}$, and every vertex $\{a,b\} \in V_i$ is joined to its two parents $a, b \in V_{i-1}$. The result is connected, bipartite and 2-degenerate by construction, and is related to the complete degenerate graphs of Grzesik, Janzer and Nagy.
The lower bound comes from a **sampled Hamming-ball graph**. With $U = \{0,1\}^m$, two disjoint copies $U_L, U_R$ are joined whenever their Hamming distance is at most $k = \lfloor \tau m\rfloor$, and each vertex is retained independently with probability $p = 2^{-\beta m}$. The two parameters are governed by the thresholds
$$A(\tau) = \kappa + \tau\log_2 3, \qquad C(\tau) = 2h(\tau) - 1,$$
and the construction needs a sampling exponent with $A(\tau) < \beta < C(\tau)$. The lower threshold controls exclusion of the layered graph; the upper one controls whether the sampled host has more than $n^{3/2}$ edges.
Exclusion runs on a conditional-entropy functional $E(u,z) = \frac{1}{m}\sum_j H(Z_j \mid X_j, Y_j)$ over parent and child arrays. An array of conditional entropy $E$ has at most $2^{mME + O(m\log_2 M)}$ realisations, while requiring its $M = \binom{L}{2}$ children to survive sampling costs $2^{-\beta mM}$ — which dominates the $2^{mL}$ possible parent arrays whenever $E < \beta$. An embedding of $H$ would therefore have to raise a bounded entropy potential by a fixed amount at each layer, which is impossible after enough layers. A second-moment argument shows the sampled graph still has $\Omega(n^{3/2+\varepsilon})$ edges, and padding extends the construction to every sufficiently large order.
The material is transplanted from the Lean 4 formalisation accompanying OpenAI's *Ten Advances in Mathematics and Theoretical Computer Science* (Chapter 10, "Counterexamples to the Compactness and Degeneracy Conjectures for Extremal Numbers", Sections 1.2 and 5–8), and re-verified in this environment: every node is proved from `[propext, Classical.choice, Quot.sound]` alone, and each staged statement's elaborated type was checked to be identical to the original declaration's. The mission is offered as a curated, closed campaign whose definitions and lemmas — binary entropy and the pair kernel, the layered construction, the Hamming-ball host and its retention measure — are reusable foundations for further work in extremal graph theory.
This is the companion result to Erdős problem #180, the Erdős–Simonovits compactness conjecture, which is formalised in the same source chapter and published as a separate mission.
3 thms1 active userReviewed
🏆Completed
Captain: Community (Bot)
Erdős Problem 180: the Erdős–Simonovits Compactness ConjectureResearch Paper
Erdős and Simonovits conjectured that forbidding a finite family of graphs cannot reduce the extremal number by more than a constant factor compared with forbidding one of its members: for every finite nonempty family $\mathcal{F}$ whose members all contain a cycle, there should be some $F \in \mathcal{F}$ and $C>0$ with $\mathrm{ex}(n,F) \le C\,\mathrm{ex}(n,\mathcal{F})$ for all large $n$. The cycle hypothesis is essential — the folklore family $\{K_{1,2}, 2K_2\}$ already defeats the original formulation — and the corrected conjecture is Erdős problem #180.
This mission carries a complete Lean 4 formalisation refuting it, and refuting it quantitatively: there is a finite family $\mathcal{F}$ of **connected bipartite** graphs, each containing a cycle, with
$$\mathrm{ex}(n,\mathcal{F}) = O\!\left(n^{4/3-1/48}\right) \qquad\text{while}\qquad \mathrm{ex}(n,F) = \Omega\!\left(n^{4/3}\right) \ \ (F \in \mathcal{F}).$$
The two bounds are separated by a polynomial factor $n^{1/48}$, so no member can dominate the family up to any constant. The family is $\mathcal{F} = \{C_4, C_6\} \cup \mathcal{J} \cup \mathcal{K}$, where $\mathcal{J}$ and $\mathcal{K}$ are the admissible quotients of two properly $2$-coloured templates built from the subdivisions of $K_{3,2}$ and $K_{3,3}$. The upper bound comes from counting short paths in an $\mathcal{F}$-free graph: excluding $\mathcal{J}$ bounds the number of vertices that fail to be centres of a subdivided $K_{3,3}$, and excluding $\mathcal{K}$ forces those vertices to form a vertex cover. The lower bound comes from incidence graphs of symplectic generalized quadrangles $W(q)$, with the characteristic of the underlying field chosen to suit the forbidden member — even $q$ for $\mathcal{J}$, odd $q$ for $\mathcal{K}$ — which is exactly the freedom a family bound does not have.
The material is transplanted from the Lean 4 formalisation accompanying OpenAI's *Ten Advances in Mathematics and Theoretical Computer Science* (Chapter 10, "Counterexamples to the Compactness and Degeneracy Conjectures for Extremal Numbers"), re-verified in this environment. Every node is proved; the mission is offered as a curated, closed campaign whose definitions and lemmas are reusable foundations for further work in extremal graph theory.
5 thms1 active userReviewed
🏆Completed
Captain: Shuze Chen
Erdős Problem 183: Multicolour Triangle Ramsey NumbersResearch Paper
How fast do multicolour Ramsey numbers grow? Write $R_k$ for the least $n$ such that every colouring of the edges of $K_n$ with $k$ colours contains a monochromatic triangle. The classical bounds, essentially unimproved for decades, place $R_k$ between $c^k$ and $e\cdot k!$, and Erdős asked repeatedly whether the truth is closer to the exponential lower end — his Problem 183 asks whether $R_k^{1/k}\to\infty$, i.e. whether the growth is genuinely superexponential.
This mission carries a complete Lean 4 formalisation resolving that question in the affirmative, with an explicit bound: $R_k \ge \left(\tfrac{1}{6e^{38}}\,k^{1/3}/\log k\right)^{k}$ for all sufficiently large $k$, from which $R_k^{1/k}\to\infty$ follows, together with the matching two-sided estimate $\log R_k = \Theta(k\log k)$ pinning the sharp coefficients. The argument is constructive: it builds triangle-free colourings by a recursive palette construction whose colour count grows fast enough to beat every exponential.
The material is transplanted from the Lean 4 formalisation accompanying OpenAI's *Ten Advances in Mathematics and Theoretical Computer Science*, re-verified in this environment. Every node is proved — the mission is offered as a curated, closed campaign whose milestones map the attack path and whose lemmas are reusable foundations for further work on multicolour Ramsey theory.
2 thms1 active userReviewed
🏆Completed
Captain: ShouqiaoWang
Erdős Problem 788: Exponent One-Half and Explicit BoundsResearch Paper
Erdős Problem 788 asks how large a set can always be retained when prescribed distinct pair-sums are forbidden. This mission formalizes the repository’s strengthened version of Theorem 1.1: an explicit lower bound valid for every $n\ge 3$, an eventual quantitative upper bound, the conclusion $f(n)=n^{1/2+o(1)}$, and the exact affirmative answer to the original upper-bound question.