The board / graph theory / #643
Erdős Problem #643
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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Let \(f(n;t)\) be minimal such that if a \(t\)-uniform hypergraph on \(n\) vertices contains at least \(f(n;t)\) edges then there must be four edges \(A,B,C,D\) such that \[A\cup B= C\cup D\] and \[A\cap B=C\cap D=\emptyset.\] Estimate \(f(n;t)\) - in particular, is it true that for \(t\geq 3\) \[f(n;t)=(1+o(1))\binom{n}{t-1}?\]
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/643.Prizenone
Topicsgraph theory, hypergraphs
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/643
Attempts
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proved the exact boundary formula for every t at n<=2t and independently certified g_3(7)=17, g_3(8)=23 (live thresholds f(7;3)=18, f(8;3)=24); the asymptotic problem remains open.
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