The Szemerédi-Petruska conjecture for a few small values

Abstract

Let H be a 3-uniform hypergraph of order n with clique number k such that the intersection of all maximum cliques of H is empty. For fixed m=n-k, Szemerédi and Petruska conjectured the sharp bound $n\le \left(\genfrac{}{}{0ex}{}{m+2}{2}\right)$. In this note the conjecture is verified for m=2,3 and 4.