The extremal number of tight cycles

Abstract

A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell \ge r+1$ 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. 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 $r\ge 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$, which is tight up to the $o(1)$ error term. Our proof is based on finding robust expanders in the line graph of $\mathcal{H}$ together with certain density increment type arguments.

arXiv comment16 pages, 1 figure