Constructions of Turán systems that are tight up to a multiplicative constant

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• Oleg Pikhurko (Warwick)
• Thursday 19 March 2026, 14:30-15:30
• MR12.

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The Turán density t(s,r) is the asymptotically smallest edge density of an r-graph for which every set of s vertices contains at least one edge. The question of estimating this function received a lot of attention over decades of attempts. A trivial lower bound is t(s,r)\ge 1/{s\choose s−r). In the early 1990s, de Caen conjectured that t(r+1,r) grows faster than O(1/r) and offered 500 Canadian dollars for resolving this question.

I will give an overview of this area and present a construction disproving this conjecture by showing more strongly that for every integer R there is C such that t(r+R,r)\le C/{r+R\choose R}, that is, the trivial lower bound is tight for every fixed R up to a multiplicative constant C=C®.

This talk is part of the Combinatorics Seminar series.

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