On tight cycles in hypergraphs

TL;DR

This paper investigates the maximum number of edges in hypergraphs without tight cycles and provides a negative answer to a longstanding conjecture, expanding understanding of hypergraph cycle structures.

Contribution

It disproves a conjecture that bounds edges in cycle-free hypergraphs, showing such bounds do not always hold.

Findings

Counterexample to the conjecture

Hypergraphs can have more edges than previously thought

Implications for hypergraph cycle theory

Abstract

A tight $k$-uniform $\ell$-cycle, denoted by $TC_\ell^k$, 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, 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 $\binom{n-1}{k-1}$ edges. In this paper, we answer this question in negative.