Covering non-uniform hypergraphs

Abstract

A subset of the vertices in a hypergraph is a cover if it intersects every edge. Let τ(H) denote the cardinality of a minimum cover in the hypergraph H, and let us denote by g(n) the maximum of τ(H) taken over all hypergraphs H with n vertices and with no two hyperedges of the same size. We show thatg(n)<1.98n(1+o(1)). A special case corresponds to an old problem of Erdos asking for the maximum number of edges in an n-vertex graph with no two cycles of the same length. Denoting this maximum by n+f(n), we can show that f(n)≤1.98n(1+o(1)). Generalizing the above, let g(n, C, k) denote the maximum of τ(H) taken over all hypergraph H with n vertices and with at most Cik edges with cardinality i for all i=1, 2, ..., n. We prove that g(n, C, k)