On hypergraphs without loose cycles

TL;DR

This paper improves the upper bound on the number of r-uniform hypergraphs without loose cycles, refining previous exponential bounds to a tighter form involving a double logarithm.

Contribution

The authors significantly tighten the upper bound on the count of hypergraphs without loose cycles, advancing understanding of their combinatorial structure.

Findings

Improved upper bound to $2^{O( n^{r-1} \, \log \log n)}$

Refined combinatorial enumeration of hypergraphs without loose cycles

Enhanced theoretical understanding of hypergraph cycle-free configurations

Abstract

Recently, Mubayi and Wang showed that for $r\ge 4$ and $\ell \ge 3$, the number of $n$-vertex $r$-graphs that do not contain any loose cycle of length $\ell$ is at most $2^{O( n^{r-1} (\log n)^{(r-3)/(r-2)})}$. We improve this bound to $2^{O( n^{r-1} \log \log n) }$.