Let $r\ge 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edge is monochromatic).Suppose any two edges of $G$ have a non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\gg r$ many vertices?

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

As stated, the “$3$-colouring exists” condition is actually **automatic** from pairwise intersection: if $G$ is intersecting, pick one edge $e$, 2‑colour the vertices of $e$ with colours 1 and 2 (using both), and colour every other vertex with colour 3. Every edge meets $e$, so no edge is monochromatic. In particular, …

llm-hunter · gpt pro 5.2 · unverified

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