Covering Non-uniform Hypergraphs

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 that g(n) < 1.98 √ n(1 + o(1)). A special case corresponds to an old problem of Erdős asking 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) ≤ 1.98√n(1 + o(1)). Generalizing the above, let g(n,C, k) denote the maximum of τ(H) taken over all hypergraphs H with n vertices and with at most Ci edges with cardinality i for all i = 1, 2, ..., n. We prove that g(n,C, k) < (Ck! + 1)n. As an interesting graph-theoretic application of these results, let us consider a (possibly infinite) family F of graphs, and 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. (Clearly, T (n, F, 1) = T (n, F ) is the classical Turán number.) The family F is said to grow polynomially if there exist a constant c > 0 and a nonnegative integer k such that for every i, there are at most ci members in F having exactly i edges. Generalizing the above mentioned problem of Erdős, we can prove that for a polynomially growing family F and for a sufficiently large n we have T (n, F, r) < T (n, F ) + (c(r − 1)k! + 1)T (n, F ) k+1 k+2 + 2(c(r − 1)k! + 1)T (n, F ) k k+2 . RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway NJ 08854-8003, U.S.A. e–mail: [email protected] Department of Mathematics, University of Haifa-ORANIM, Tivon 36006, Israel. e–mail: [email protected] Mathematical Institute of the Hungarian Academy of Sciences, P.O.B. 127, Budapest 1364 Hungary. e–mail: [email protected] Department of Mathematics, University of Haifa-ORANIM, Tivon 36006, Israel. e–mail: [email protected]