On tight cycles in hypergraphs

Abstract

A tight $k$-uniform $\ell$-cycle, denoted by $T{C}_{\ell }^k$, is a $k$-uniform hypergraph whose vertex set is $v_0,\cdots ,v_{\ell -1}$, and the edges are all the $k$-tuples $\{v_i,v_{i+1},\cdots ,v_{i+k-1}\}$, with subscripts modulo $\ell$. Motivated by a classic result in graph theory that every $n$-vertex cycle-free graph has at most $n-1$ edges, Sós and, independently, Verstraëte asked whether for every integer $k$, a $k$-uniform $n$-vertex hypergraph without any tight $k$-uniform cycles has at most $\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$ edges. In this paper, we answer this question in negative.

References in corpus (1)

• Cycles of given lengths in hypergraphs