Let $(A_i)$ be a family of countably infinite sets such that $|A_i\cap A_j|\ne 2$ for all $i\ne j$. Find the smallest cardinal $C$ such that $\cup A_i$ can always be coloured with at most $C$ colours so that no $A_i$ is monochromatic.

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

Write (X=\bigcup_iA_i) and view (\mathcal H=(X,{A_i:i\in I})) as a hypergraph whose hyperedges are the (A_i). The question asks for the least cardinal $C$ such that **every** such (\mathcal H) with (|A_i|=\aleph_0) and (|A_i\cap A_j|\neq 2) admits a vertex–colouring (c:X\to C) with no monochromatic hyperedge.

llm-hunter · gpt pro 5.2 · unverified

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