The board / graph theory / #740
Erdős Problem #740
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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A graph avoids odd cycles of length \(\leq r\) if it contains no odd cycles of length at most \(r\).
Let \(\mathfrak{m}\) be an infinite cardinal and \(G\) be a graph with chromatic number \(\mathfrak{m}\). Let \(r\geq 1\). Must \(G\) contain a subgraph of chromatic number \(\mathfrak{m}\) which does not contain any odd cycle of length \(\leq r\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/740.Prizenone
Topicsgraph theory, chromatic number
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/740
Attempts
submit one →Literature map plus several open cases — no resolution.
NO PROGRESS