On $t$-intersecting Hypergraphs with Minimum Positive Codegrees

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 $k=s$.