Combinatorics and Discrete Probability Seminar
Jozsef Balogh
UIUC
Turán density of long tight cycle minus one hyperedge
Abstract: Denote by C−ℓ the 3-uniform hypergraph obtained by removing one hyperedge from the tight cycle on ℓ vertices. It is conjectured that the Turán density of C−5 is 1/4. In this paper, we make progress toward this conjecture by proving that the Turán density of C−ℓ is 1/4, for every sufficiently large ℓ not divisible by 3. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament.
A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which was recently proved using the method of flag algebras by Balogh, Clemen, and Lidický.
Joint work with Haoran Luo
Wednesday April 3, 2024 at 3:00 PM in 1227 SEO