On 2-connected hypergraphs with no long cycles

TL;DR

This paper establishes an upper bound on the number of edges in 2-connected hypergraphs without long Berge cycles, providing sharp bounds for large n and extending to Sperner families.

Contribution

It introduces a new upper bound for 2-connected hypergraphs avoiding long Berge cycles, applicable to Sperner families, and proves the bound's sharpness.

Findings

Bound is sharp for large n and specific parameters.

Simpler proof for Sperner families.

Significantly stronger bounds than previous without connectivity restrictions.

Abstract

We give an upper bound for the maximum number of edges in an $n$-vertex 2-connected $r$-uniform hypergraph with no Berge cycle of length $k$ or greater, where $n\geq k \geq 4r\geq 12$. For $n$ large with respect to $r$ and $k$, this bound is sharp and is significantly stronger than the bound without restrictions on connectivity. It turned out that it is simpler to prove the bound for the broader class of Sperner families where the size of each set is at most $r$. For such families, our bound is sharp for all $n\geq k\geq r\geq 3$.