GPT 6 Astra Ultra attempt on Math-5401

Erdős Problem #61: independent continuation

Date: 2026-09-23. Research attempt; no claim of a new published result.

1) TARGET AND SOURCE AUDIT

For every fixed graph $H$, does every $H$-free graph on $n$ vertices have a clique or independent set of size at least $n^{\delta(H)}$ for some $\delta(H)>0$?

All supplied versions considered: 61.tex. The source reduces to prime forbidden graphs using an external substitution theorem and obtains upper bounds from repeated identical substitution. I verify a further quantitative obstruction using alternating a graph and its complement; the argument below is conditional only on both bases being $H$-free.

2) WORK

Suppose $H$ is prime and both $B$ and $\overline B$ are $H$-free. Write $b=|B|$, $a=\alpha(B)$, $w=\omega(B)$. Form an alternating lexicographic product with $2t$ factors, $t$ copies of each. Then \[ |G_t|=b^{2t},\qquad \alpha(G_t)=\omega(G_t)=(aw)^t. \] For completeness, an independent set in $X[Y]$ meets an independent set of blocks of $X$ and takes at most $\alpha(Y)$ vertices in each; choosing maximum sets attains the product. The clique argument is identical. In the complement the two parameters exchange.

These products remain $H$-free: in an induced copy of a prime $H$, the intersection with any substituted block is a module; hence either the copy is wholly in a single block or it uses at most one vertex from each block. Either option would give an induced $H$ in one of the factors. It follows that any valid exponent satisfies \[ \delta(H)\le {\log(aw)\over2\log b}. \] This is at least as strong as $\log_b\max(a,w)$, and strictly stronger when $a\ne w$. The extra condition on $\overline B$ is automatic if $b<|H|$, and also when $H\cong\overline H$ and $B$ is $H$-free.

3) ADVERSARIAL VERIFICATION

If $H$ is not self-complementary and $b\ge|H|$, the complement need not be $H$-free, so the improvement cannot be used automatically. Multiplication concerns the separate clique and independence parameters, not their maximum at intermediate levels. This supplies upper obstructions, not the positive lower bound required by the conjecture.

4) FINAL

UNRESOLVED. The general prime forbidden-graph case remains unresolved. No new positive Erdős--Hajnal exponent is established here.

5) SOURCES CHECKED

No external theorem is used in the new argument. The supplied source was read in full; the saved Tao status snapshot is dated 2026-09-22.

6) COMPLETION ESTIMATE

This is a conservative subjective measure of progress toward the original question, not a probability of correctness or a claim that the remaining work is routine. COMPLETION: 8%