On 3-uniform hypergraphs without a cycle of a given length

TL;DR

This paper investigates the maximum size of 3-uniform hypergraphs without certain cycle lengths, providing improved upper bounds for the number of hyperedges in such hypergraphs, especially for odd cycle lengths.

Contribution

The paper presents new upper bounds on the number of hyperedges in 3-uniform hypergraphs avoiding specific Berge cycles, improving previous bounds by a factor of Θ(k^2).

Findings

Upper bound for C_{2k+1}-free hypergraphs is O(k^2 n^{1+1/k}.

Improved bounds for linear hypergraphs avoiding certain cycles.

Enhanced understanding of cycle-free hypergraph extremal sizes.

Abstract

We study the maximum number of hyperedges in a 3-uniform hypergraph on $n$ vertices that does not contain a Berge cycle of a given length $\ell$. In particular we prove that the upper bound for $C_{2k+1}$-free hypergraphs is of the order $O(k^2n^{1+1/k})$, improving the upper bound of Gy\"ori and Lemons by a factor of $\Theta(k^2)$. Similar bounds are shown for linear hypergraphs.