On a conjecture of Szemer\'edi and Petruska

Consider a $3$-uniform hypergraph of order $n$ with clique number $k$ such that the intersection of all its $k$-cliques is empty. Szemer\'edi and Petruska proved $n\leq 8m^2+3m$, for fixed $m=n-k$, and they conjectured the sharp bound $n\leq{m+2\choose 2}$. Tuza proved the best known bound, $n\leq \frac{3}{4}m^2+m+1$, using the machinery of $\tau$-critical hypergraphs. Here we propose an alternative approach, combining a decomposition process introduced by Szemer\'edi and Petruska with the skew version of Bollob\'as's theorem to prove $n\leq m^2 + 6m + 2$. While the bound obtained here is weaker than Tuza's bound, it is a proof-of-concept for a different approach and a call to apply dimension bounds from linear algebra.