# $3$-uniform hypergraphs without a cycle of length five

## TL;DR

This paper establishes an improved upper bound on the maximum number of hyperedges in a 3-uniform hypergraph on n vertices that avoids cycles of length five, advancing understanding of hypergraph cycle constraints.

## Contribution

It provides a tighter upper bound on hyperedges in 3-uniform hypergraphs without 5-cycles, refining previous estimates by Bollobás and Győri.

## Key findings

- Maximum hyperedges less than (0.254 + o(1))n^{3/2}
- Fewer 3-paths originate from certain subgraphs of the shadow
- Improved bounds on cycle-free hypergraph configurations

## Abstract

In this paper we show that the maximum number of hyperedges in a $3$-uniform hypergraph on $n$ vertices without a (Berge) cycle of length five is less than $(0.254 + o(1))n^{3/2}$, improving an estimate of Bollob\'as and Gy\H{o}ri. We obtain this result by showing that not many $3$-paths can start from certain subgraphs of the shadow.