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Erdős #500 · graph theory / hypergraphs

What is ex3(n,K43)\mathrm{ex}_3(n,K_4^3)? That is, the largest number of 33-edges which can placed on nn vertices so that there exists no K43K_4^3, a set of 4 vertices which is covered by all 4 possible 33-edges.

MSC 05C · Graph theory

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

What is ex3(n,K43)\mathrm{ex}_3(n,K_4^3)? That is, the largest number of 33-edges which can placed on nn vertices so that there exists no K43K_4^3, a set of 4 vertices which is covered by all 4 possible 33-edges.

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Background

A problem of Turán. Turán observed that dividing the vertices into three equal parts X1,X2,X3X_1,X_2,X_3, and taking the edges to be those triples that either have exactly one vertex in each part or two vertices in XiX_i and one vertex in Xi+1X_{i+1} (where X4=X1X_4=X_1) shows that

ex3(n,K43)≥(59+o(1))(n3).\mathrm{ex}_3(n,K_4^3)\geq\left(\frac{5}{9}+o(1)\right)\binom{n}{3}.

This is probably the truth. The current best upper bound is

ex3(n,K43)≤0.5611666(n3),\mathrm{ex}_3(n,K_4^3)\leq 0.5611666\binom{n}{3},

due to Razborov [Ra10].

See also [712] for the general case.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #500 link

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