Dear all,

There will be a noon seminar on TUE(!)sday, 16 July, given by Jozsef Balogh. Please find the talk details below.

Best regards,
Misha Tyomkyn.
Turán density of long tight cycle minus one hyperedge
Jozsef Balogh
University of Illinois Urbana-Champaign
July 16, 2024, 12:20 in S6

Abstract
Denote by $C_\ell^-$ the $3$-uniform hypergraph obtained by removing one hyperedge from the tight cycle on $\ell$ vertices. It is conjectured that the Tur\'a density of $C_5^-$ is $1/4$. In this paper, we make progress toward this conjecture by proving that the Tur\'an density of $C_\ell^-$ is $1/4$, for every sufficiently large $\ell$ 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 Lidick\'y.

Joint work with Haoran Luo.
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