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.
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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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