AI-Assisted Proof Disproves Erdős's Degeneracy Conjecture in Graph Theory
ctjlewis · x · 2026-08-02
A major breakthrough in AI-assisted mathematical proof has successfully disproved the famous Erdős's degeneracy conjecture.
The conjecture posited that for an $r$-degenerate bipartite graph $H$, its extremal function satisfies $ex(n;H) \ll n^{2-1/r}$. However, the new proof demonstrates that this fails for every $r \ge 2$: for each $r$, there exists an $r$-degenerate bipartite graph $Hr$ such that $ex(n;Hr) \ge n^{2-1/r+\epsilonr}$ (where $\epsilonr \sim 1/(16r^2)$). This result is derived via an exact $1/r$ cancellation between binomial variance and entropy curvature at a specific Gibbs weight and Hamming radius. For instance, at $r=3$, $ex(n;H) \ge n^{5/3+1/200}$.
Related event: AI-Assisted Proof Refutes Classic Erdős Degeneracy Conjecture(2 posts)→
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