The board / graph theory / #500
Erdős Problem #500
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).
What is \(\mathrm{ex}_3(n,K_4^3)\)? That is, the largest number of \(3\)-edges which can placed on \(n\) vertices so that there exists no \(K_4^3\), a set of 4 vertices which is covered by all 4 possible \(3\)-edges. See also [712](https://www.erdosproblems.com/712) for the general case.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/500.Prize$500
Topicsgraph theory, hypergraphs, turan number
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA140462
Original pageerdosproblems.com/500
Attempts
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independently proved and checked the exact table through n=9 and the sharp 5/9 bound for every three-colour-multiset blow-up; the unrestricted density gap remains, and the live page's 0.5611666 decima
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