Hypergraphs with no tight cycles

TL;DR

This paper establishes an improved upper bound on the number of edges in r-uniform hypergraphs without tight cycles, nearly matching a recent construction, thus advancing understanding of hypergraph extremal properties.

Contribution

The paper improves the upper bound on edges in hypergraphs without tight cycles, refining previous results and nearly matching known constructions.

Findings

New upper bound of O(n^{r-1} (log n)^5) edges

Improved upon previous bound of n^{r-1} e^{O(√log n)}

Bound is tight up to a polylogarithmic factor

Abstract

We show that every $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle has at most $O(n^{r-1} (\log n)^5)$ edges. This is an improvement on the previously best-known bound, of $n^{r-1} e^{O(\sqrt{\log n})}$, due to Sudakov and Tomon, and our proof builds up on their work. A recent construction of B. Janzer implies that our bound is tight up to an $O((\log n)^4 \log \log n)$ factor.