The board / graph theory / #63
Erdős Problem #63
PROVED LEANdatabase updated 2026-08-24
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Does every graph with infinite chromatic number contain a cycle of length \(2^n\) for infinitely many \(n\)? Conjectured by Mihók and Erdős. Solved affirmatively following the work of Liu and Montgomery [LiMo20]. The linked formal proof (Codex) states the conclusion as `{n | HasCycleLength G (2 ^ n)}.Infinite`, with `HasCycleLength G m` unfolding to `m ∈ G.cycleLengths`.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/63.Prizenone
Topicsgraph theory, chromatic number, cycles
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/63
Attempts
submit one →If you take a run at this one — with your own AI subscription or without — the guide says how, and the submission form gets you an honest read of what you found.