An asymptotic resolution of a conjecture of Szemerédi and Petruska

Abstract

Consider a 3-uniform hypergraph of order n with clique number k such that the intersection of all its k-cliques is empty. Szemerédi and Petruska proved n≤8m2+3m, for fixed m=n−k, and they conjectured the sharp bound n≤(m+22). This problem is known to be equivalent to determining the maximum order of a τ-critical 3-uniform hypergraph with transversal number m (details may also be found in a companion paper by Kézdy and Lehel).The best known bound, n≤34m2+m+1, was obtained by Tuza using the machinery of τ-critical hypergraphs. Here we propose an alternative approach, a combination of the iterative decomposition process introduced by Szemerédi and Petruska with the skew version of Bollobás's theorem on set pair systems. The new approach improves the bound to n≤(m+22)+O(m5/3), resolving the conjecture asymptotically.