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.