On $t$-intersecting Hypergraphs with Minimum Positive Codegrees

TL;DR

This paper characterizes the maximum size of certain intersecting hypergraphs with positive codegree constraints, generalizing classical theorems and identifying unique extremal structures for large n.

Contribution

It extends previous results by determining the extremal hypergraphs with minimum positive codegree conditions for a broad range of parameters.

Findings

Identifies the unique maximum-edge hypergraph structure under given conditions.

Generalizes the Erd ext{"o}s-Ko-Rado theorem for new parameter ranges.

Provides bounds and conditions for large n to ensure extremality.

Abstract

For a hypergraph $\mathcal{H}$, define the minimum positive codegree $\delta_i^+(\mathcal{H})$ to be the largest integer $k$ such that every $i$-set which is contained in at least one edge of $\mathcal{H}$ is contained in at least $k$ edges. For $1\le s\le k,t$ and $t\le r$, we prove that for $n$-vertex $t$-intersecting $r$-graphs $\mathcal{H}$ with $\delta_{r-s}^+(\mathcal{H})>{k-1\choose s}$, the unique hypergraph with the maximum number of edges is the hypergraph $\mathcal{H}$ consisting of every edge which intersects a set of size $2k-2s+t$ in at least $k-s+t$ vertices provided $n$ is sufficiently large. This generalizes work of Balogh, Lemons, and Palmer who proved this for $s=t=1$, as well as the Erd\H{o}s-Ko-Rado theorem when $k=s$.