Covering Non-uniform Hypergraphs

A subset of the vertices in a hypergraph is a cover if it intersects every edge. Let ?(H) denote the cardinality of a minimum cover in the hypergraph H, and let us denote by g(n) the maximum of ?(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 Erdo?s 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 ?(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)<(Ck!+1)n(k+1)/(k+2).These results have an interesting graph-theoretic application. For a family F of graphs, let T(n, F, r) denote the maximum possible number of edges in a graph with n vertices, which contains each member of F at most r?1 times. T(n, F, 1)=T(n, F) is the classical Turan number. Using the results above, we can compute a non-trivial upper bound for T(n, F, r) for many interesting graph families.