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.
Categories: math.CO