The board / set theory / #1174
Erdős Problem #1174
NOT DISPROVABLEdatabase updated 2026-03-19
Open in general, but there are models of set theory in which it is true — so it can't be disproved from the usual axioms (ZFC) alone.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
**Erdős Problem 1174, part (i)** [Va99, 7.91]. A problem of Erdős and Hajnal: does there exist a graph \(G\) with no \(K_4\) such that every edge colouring of \(G\) with countably many colours contains a monochromatic \(K_3\)? For simplicity the colourings are given on all unordered pairs \(\operatorname{Sym}_2(V)\); only the values on the edges of \(G\) matter, since a monochromatic triangle is witnessed by edges alone. Shelah proved that a graph with this property can consistently exist, so the problem is not disprovable in ZFC; whether such a graph exists outright remains open.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1174.Prizenone
Topicsset theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1174
Attempts
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