Let $k\ge 3$. Is it true that, if $m$ is sufficiently large, for any graph $H$ on $m$ edges without isolated vertices,$$R(C_k,H)\le 2m+\lfloor \frac{k-1}{2}\rfloor ?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Take (H=K_2) (one edge, no isolated vertices), so $m=1$. Then [ R(C_k,K_2)=k. ] Reason: on (K_{k-1}) color every edge red; there is no red (C_k) (too few vertices) and no blue (K_2) (no blue edges), so (R(C_k,K_2)\ge k). On (K_k), if there is any blue edge we already have a blue (K_2); if not, all edges are red and (K_…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.