The board / graph theory / #57
Erdős Problem #57
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).
If \(G\) is a graph with infinite chromatic number and \(a_1 < a_2 < \cdots\) are lengths of the odd cycles of \(G\) then \(\sum \frac{1}{a_i} = \infty\). Conjectured by Erdős and Hajnal [ErHa66], and solved by Liu and Montgomery [LiMo20]. The linked formal proof (Codex and GPT-5.6 Sol) states the conclusion as `¬ Summable (oddCycleReciprocal G)`, where `oddCycleReciprocal G n` is `n⁻¹` if `n` is an odd cycle length of `G` and `0` otherwise; this is the indicator form of the sum below.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/57.Prizenone
Topicsgraph theory, chromatic number, cycles
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/57
Attempts
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