Claim in Erdős problems: literature check, graph theory (1)
A published paper or arXiv preprint answers the main question of Erdős Problem #108 (theorem erdos_108 in formal-conjectures) in full (proof, disproof, or the exact value asked for) with no hypotheses beyond the problem's own.
For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r) contains a subgraph of girth ≥ r and chromatic number ≥ k?
Verdict: candidate reference, not checked by a mathematician
Evidence for
“For every k≥1, there exists a triangle-free graph with chromatic number greater than k whose C4-free subgraphs are all 3-colourable. In particular, this implies that ℓ(5,k) is infinite for all k≥3.”
Note: T. Nguyen, B. Walczak (2026), arXiv:2609.40192 (v2 1 Oct 2026; expository, unrefereed), Thm 1.4, proved in the note. Match: their Problem 1.1 is erdos_108 with k shifted by 1; in a triangle-free graph a girth>=5 subgraph is exactly a C4-free one, so r=5, k=4 has no finite f: erdos_108 is false (answer no). Builds on Kohlmeyer-Kruer 2026 manuscripts (not on arXiv).
Evidence against
None yet.