Degree versions of the Erdős-Ko-Rado theorem and Erdős hypergraph matching conjecture

We use an algebraic method to prove a degree version of the celebrated Erdős-Ko-Rado theorem: given n > 2k, every intersecting k-uniform hypergraph H on n vertices contains a vertex that lies on at most ( n−2 k−2 ) edges. This result implies the Erdős-Ko-Rado Theorem as a corollary. It can also 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 Erdős from 1976, we show that, given integers n, k, s with n ≥ 3ks, every k-uniform hypergraph H on n vertices with minimum vertex degree greater than ( n−1 k−1 ) − ( n−s k−1 ) contains s disjoint edges.