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Updated Oct 8

Problem 594

claims/: The 2 claim pages of Problem 594, one per claimant's result; the problem's standing derives from them.

Statement. Does every graph $G$ with chromatic number $\geq \aleph_1$ contain all sufficiently large odd cycles?

Status. PROVED (LEAN).

Source. erdosproblems.com/594, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #594, https://www.erdosproblems.com/594.

References.

[EHS74] Erdős, P. and Hajnal, A. and Shelah, S., On some general properties of chromatic numbers. Topics in topology (Proc. Colloq., Keszthely, 1972) (1974), 243-255.

Formalization. Statement in formal-conjectures, which marks it research solved with a formal-proof link to Boris Alexeev's repository, recorded on the Erdős–Hajnal–Shelah claim page; nothing was built or audited here.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (14)

Problem 737 Thomassen finds an edge on cycles of all large lengths Large odd circuits announced for chromatic number above omega_2 Uncountable chromatic number forces all large odd cycles problems/set_theory/E0594/claims Set Theory and Infinite Combinatorics graph_coloring/erdos_1974_general_properties_chromatic_numbers Theorem 3 (p. 251): uncountable chromatic number forces all sufficiently long odd circuits ramsey_theory/erdos_1975_problems_results_finite_infinite_graphs Problem (Section II, p. 184): cycles of all large lengths through one edge in an uncountably chromatic graph Set Theory and Infinite Combinatorics set_theory/erdos_1966_chromatic_number_graphs_set_systems Problem 7.6: all long odd circuits at chromatic number above ω Theorem 7.5: infinite chromatic number forces infinitely many odd circuit lengths

Graph