The Szemerédi–Petruska conjecture for a few small values

Abstract Let H be a 3-uniform hypergraph of order n with clique number $$\omega (H)=k$$ ω ( H ) = k . Assume that the union k -cliques equals its vertex set, intersection all maximum cliques is empty, but one -clique non-empty. For fixed $$m=n-k$$ m n - , Szemerédi and Petruska conjectured sharp bound $$n\hbox {\,\,\char 054\,\,}{m+2\atopwithdelims ()2}$$ 6 + 2 In this note conjecture verified for $$m=2,3$$ , 3 4.