GPT 6 Astra Ultra attempt on Math-926

Problem 766: the two possible families immediately below the endpoint

1) OBJECTIVE AND SOURCE REVIEW

Attempt dated 2026-09-23. I read the complete 766.tex. It defines $f(n;k,l)=\min_{|V(H)|=k,|E(H)|=l}\operatorname{ex}(n;H)$ and leaves the strict monotonicity question in $l$ unresolved. Its eventual bipartite reduction and balanced endpoint do not evaluate the underlying extremal numbers. I classify the candidates one edge below that endpoint and simplify the reduction without an Erdős--Stone input.

2) WORK

Every nonbipartite $H$ is avoided by the balanced complete bipartite graph, so \[ \operatorname{ex}(n;H)\ge\lfloor n^2/4\rfloor \] uniformly, without a graph-dependent threshold. There is a bipartite candidate $B$ whenever $l\le\lfloor k^2/4\rfloor$, and $B\subseteq K_{k,k}$. An elementary common-neighbour count gives $\operatorname{ex}(n;K_{k,k})=O_k(n^{2-1/k})$: in a $K_{k,k}$-free graph, \[ \sum_v\binom{d(v)}k\le(k-1)\binom nk. \] Indeed every $k$-vertex set has at most $k-1$ common neighbours. The inequality $\binom dk\ge(d-k+1)_+^k/k!$ and convexity bound the average degree by $(k-1)^{1/k}n^{1-1/k}+k-1$. Thus the fixed bipartite candidate eventually beats every nonbipartite candidate simultaneously. This reproves the needed reduction directly.

Let $k=2r$, $r\ge3$, and $l=r^2-1$. In a bipartite candidate with parts of sizes $a,2r-a$, we need \[ a(2r-a)=r^2-(a-r)^2\ge r^2-1. \] Up to interchanging the sides, either $a=r$ or $a=r-1$. The former graph is $K_{r,r}$ with one edge removed; the latter is the full $K_{r-1,r+1}$. Consequently, for all sufficiently large $n$, \[ f(n;2r,r^2-1)=\min\{\operatorname{ex}(n;K_{r,r}-e), \operatorname{ex}(n;K_{r-1,r+1})\}. \] For $k=2r+1$, $r\ge3$, one edge below the maximum means $l=r(r+1)-1$. An unbalanced split has at most $(r-1)(r+2)=r(r+1)-2$ edges. Only the balanced split survives, giving \[ f(n;2r+1,r(r+1)-1)=\operatorname{ex}(n;K_{r,r+1}-e) \] eventually. Edge transitivity makes the choice of deleted edge immaterial.

3) ADVERSARIAL VERIFICATION

These are exact identifications with other extremal functions, not numerical evaluations of those functions. Isolated vertices do not create extra candidates: every bipartition must still have capacity at least $l$, and the near-maximum edge count forces the listed graphs. Ordinary non-strict monotonicity follows by deleting an edge from an $(l+1)$-edge minimizer; strict inequality does not follow from graph containment alone.

4) FINAL

UNRESOLVED. The first layer below the endpoint reduces to one or two explicit extremal numbers. Their necessary strict comparisons, and the general strict-monotonicity question, remain unproved.

5) SOURCES CHECKED

The complete supplied 766.tex, read 2026-09-23. The bipartite reduction, common-neighbour estimate, and near-endpoint classification are proved here and require no uninspected external theorem.

6) COMPLETION ESTIMATE

Subjective progress toward the strict-monotonicity problem. COMPLETION: 15%