Erdős Problem #638 — Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\in S$ such that if the edges of $G_n$ are coloured with $n$ colours then there is a monochromatic triangle.

Problem $638$ asks whether finite triangle-Ramsey richness of a family $S$ forces, for every infinite cardinal $aleph$, a graph whose finite subgraphs lie in $S$ and which is triangle-Ramsey for $aleph$ edge-colours. The page records no proposer or date.

Known results

• Closure under arbitrary subgraphs is insufficient: a sparse family of complete graphs gives a counterexample.
• The intended formulation appears to require closure under finite subgraphs; substantial related literature is cited, but no named theorem is supplied in the retrieved material.
• A Lean project formalizes only the compactness and diagonal reduction; its finite-avoidance and block-sequence lemmas remain unproved.

Reported construction

A newer discussion says that a construction addresses the hereditary finite-subgraph interpretation, while treating the induced-subgraph analogue separately. No author, date, construction details, or complete proof are provided, so this is unverified progress rather than a settled solution.

Current status (as of March 2026): the original formulation has a counterexample; the intended hereditary version has only an undocumented reported construction and remains unverified.