The board / graph theory / #564
Erdős Problem #564
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(R_3(n)\) be the minimal \(m\) such that if the edges of the \(3\)-uniform hypergraph on \(m\) vertices are \(2\)-coloured then there is a monochromatic copy of the complete \(3\)-uniform hypergraph on \(n\) vertices. Is there some constant \(c>0\) such that \[ R_3(n) \geq 2^{2^{cn}}? \]
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/564.Prize$500
Topicsgraph theory, ramsey theory, hypergraphs
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/564
Attempts
submit one →Read our working report →
proved a uniform barrier excluding every fixed pointwise two-colour merge of the standard four-colour stepping-up construction, and independently verified the known explicit 87-vertex colouring \(R_3(
PARTIAL