The board / graph theory / #640
Erdős Problem #640
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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An odd cycle whose vertices induce a graph of chromatic number at least `k`.
For every \(k \geq 3\), does there exist \(f(k)\) such that every finite graph of chromatic number at least \(f(k)\) has an odd cycle whose vertices induce a graph of chromatic number at least \(k\)? The cycle may have chords in the ambient graph.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/640.Prizenone
Topicsgraph theory, chromatic number
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/640
Attempts
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Exact catalogue-backed verification proves the \(k=4\) assertion for every graph on at most 12 vertices, with sharp threshold \(\chi=5\), and determines the unique smallest path-version witness at ord
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