On tight cycles in hypergraphs

A tight $k$-uniform $\mathrm{\ell }$-cycle, denoted by $T{C}_{\mathrm{\ell }}^k$, is a $k$-uniform hypergraph whose vertex set is $v_0,\cdots ,v_{\mathrm{\ell }-1}$, and the edges are all the $k$-tuples $\{v_i,v_{i+1},\cdots ,v_{i+k-1}\}$, with subscripts modulo $\mathrm{\ell }$. Motivated by a classic result in graph theory that every $n$-vertex cycle-free graph has at most $n-1$ edges, S'os and, independently, Verstra"ete asked whether for every integer $k$, a $k$-uniform $n$-vertex hypergraph without any tight $k$-uniform cycles has at most $\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)$ edges. In this paper, we answer this question in negative.