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Does every graph with infinite chromatic number contain a cycle of length for infinitely many ?
Source: erdosproblems.com/63
An accepted solution exists. The statement is true.
The site labels the problem PROVED (LEAN) and credits Zach Hunter with deducing it from Liu and Montgomery's even-cycle interval theorem; the Lean behind the qualifier is described below. The accepted full claim is powers of two from the even-cycle interval theorem; the uncountable-chromatic case is an accepted partial claim, every power of two at uncountable chromatic number.