On Ryser’s conjecture

Abstract

Motivated by an old problem known as Ryser’s Conjecture, we prove that for r = 4 and r = 5, there exists ɛ> 0 such that every r-partite r-uniform hypergraph H has a cover of size at most (r − ɛ)ν(H), where ν(H) denotes the size of a largest matching in H.