MATH Seminar

Title: The Extremal Number of Tight Cycles
Seminar: Combinatorics
Speaker: Istvan Tomon of ETH Zurich
Contact: Dr. Hao Huang, hao.huang@emory.edu
Date: 2020-10-02 at 10:00AM
Venue: https://emory.zoom.us/j/96323787117
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Abstract: A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell\geq r+1$ vertices $x_1,\dots,x_{\ell}$ such that all $r$-tuples $\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$ (with subscripts modulo $\ell$) are edges of $\mathcal{H}$. An old problem of V. S\'os, also posed independently by J. Verstra\"ete, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\geq 3$. In my talk, I will present a brief outline of the proof of the upper bound $n^{r-1+o(1)}$, which is tight up to the $o(1)$ error term. This is based on a joint work with Benny Sudakov.