The extremal number of tight cycles

TL;DR

This paper determines the maximum number of edges in an r-uniform hypergraph on n vertices without tight cycles, establishing an asymptotically tight upper bound of n^{r-1+o(1)} using expansion and density techniques.

Contribution

It provides the first near-optimal upper bound for the extremal number of tight cycles in hypergraphs, resolving a longstanding open problem.

Findings

Maximum edges in hypergraphs without tight cycles is at most n^{r-1+o(1)}

Established methods involve robust expanders and density increment arguments

Result is tight up to the o(1) term

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 $r\geq 3$. 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.