Degree versions of the Erdos-Ko-Rado Theorem and Erdos hypergraph matching conjecture

Abstract

We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\left(\genfrac{}{}{0ex}{}{n-2}{k-2}\right)$ edges. This result can be viewed as a special case of the degree version of a well-known conjecture of Erdős on hypergraph matchings. Improving the work of Bollobás, Daykin, and Erd\H os from 1976, we show that given integers $n,k,s$ with $n\ge 3k^{2}s$, every $k$-uniform hypergraph $H$ on $n$ vertices with minimum vertex degree greater than $\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)-\left(\genfrac{}{}{0ex}{}{n-s}{k-1}\right)$ contains $s$ disjoint edges.