Erdős Problem #603 — Let $(A_i)$ be a family of countably infinite sets such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$.

The problem asks for the least uniform colouring bound preventing every member of a family of countably infinite sets from being monochromatic when pairwise intersections avoid size $2$. The discussion distinguishes arbitrary-sized families from families indexed by a countable set.

Known results

• Komjáth showed that countably many colours suffice under the related condition that pairwise intersections are not of size $1$.

April–July 2026 claimed counterexample

A forum discussion attributes a construction to GPT 5.4 Pro, prompted by Przemek Chojecki, using countably infinite complete subgraphs and the Erdős–Rado theorem. It claims that arbitrary-sized families admit no uniform set-sized colour bound, while the countable-family interpretation has exact answer $2$. The construction has no independent verification or published proof in the retrieved sources.

Current status (as of September 2026): The claimed arbitrary-family refutation and the claimed countable-family answer $2$ remain unverified; Komjáth's countable-colour upper bound under the related intersection condition is the only recorded classical result.