SOLVED

Does every graph with infinite chromatic number contain a cycle of length $2^n$ for infinitely many $n$?

Conjectured by Mihók and Erdős. It is likely that $2^n$ can be replaced by any sufficiently quickly growing sequence (e.g. the squares).

David Penman has observed that this is certainly true if the graph has uncountable chromatic number, since by a result of Erdős and Hajnal [ErHa66] such a graph must contain arbitrarily large finite complete bipartite graphs (see also Theorem 3.17 of Reiher [Re24]).

Zach Hunter has observed that this follows from the work of Liu and Montgomery [LiMo20]: if $G$ has infinite chromatic number then, for infinitely many $r$, it must contain some finite connected subgraph $G_r$ with chromatic number $r$ (via the de Bruijn-Erdős theorem [dBEr51]). Each $G_r$ contains some subgraph $H_r$ with minimum degree at least $r-1$, and hence via Theorem 1.1 of [LiMo20] there exists some $\ell_r\geq r^{1-o(1)}$ such that $H_r$ contains a cycle of every even length in $[(\log \ell)^8,\ell]$.

See also [64].