The Extremal Number of Tight Cycles

Abstract

Abstract A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell \geq r+1$ vertices $x_1,...,x_{\ell }$ such that all $r$-tuples $\{x_{i},x_{i+1},...,x_{i+r-1}\}$ (with subscripts modulo $\ell $) are edges of $\mathcal{H}$. An old problem of V. Sós, also posed independently by J. Verstraëte, 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 $r\geq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$. This is tight up to the $o(1)$ error term, and it was shown recently by B. Janzer that this error term is indeed needed. Our proof is based on finding robust expanders in the line-graph of $\mathcal{H}$ together with certain density increment type arguments.