Disproved on the site (label DISPROVED; page last edited 1 October 2025). The site attributes the conjecture to Erdős, Hajnal and Szemerédi [EHS82], notes that it fails for graphs of chromatic number $\aleph_0$, recalls that de Bruijn and Erdős [dBEr51] guarantee a finite subgraph of every finite chromatic number inside any graph of infinite chromatic number, records Erdős's view in [Er95d] that the answer should be yes with an $F$ growing faster than every iterated exponential, and credits Shelah [KoSh05] (a paper joint with Komjáth) with the consistency of a negative answer and Lambie-Hanson [La20] with a counterexample in ZFC. The standing rests on Lambie-Hanson's theorem, accepted on its refereed publication and the curator's credit: for every function $f$ there is a graph of chromatic number $\aleph_1$ in which, for every $k\ge3$, every subgraph of chromatic number at least $k$ has at least $f(k)$ vertices, so no $F$ works. The Komjáth–Shelah forcing result is an accepted partial claim: it showed the positive answer unprovable in ZFC without deciding the question.