Hypergraphs without Subgraphs of Given Connectivity

TL;DR

This paper determines the maximum number of edges in large hypergraphs without $(k+1)$-connected subgraphs, extending classical graph theory results to hypergraphs with new bounds and methods.

Contribution

It provides the first asymptotic bounds for hypergraphs avoiding $(k+1)$-connected subgraphs, using novel combinatorial and optimization techniques.

Findings

Established the maximum edge count up to an $O(n)$ error term for all $r \\ge 3$.

Provided tight bounds for hypergraphs with no $(k+1)$-connected subgraph on more than $Ck$ vertices.

Extended classical graph connectivity results to hypergraphs with new bounds and methods.

Abstract

In this paper, we study the problem of determining the maximum number of edges in an $n$-vertex $r$-uniform hypergraph that contains no $(k+1)$-connected subgraph. The graph case is a classical problem initiated by Mader, central to graph theory, and still open. First, for all $r \ge 3$, we determine this maximum up to an $O(n)$ error term, thereby identifying its leading term. We also address a related question of Carmesin by establishing a tight bound for $r$-uniform hypergraphs with no $(k+1)$-connected subgraph on more than $Ck$ vertices for any constant $C>2$ and sufficiently large $r$, and further obtain an asymptotically tight bound in the case $C=2$. Our proof combines the separator tree method introduced by Carmesin with several new combinatorial and optimization techniques, and we conclude with related remarks and open problems.