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 $n\le 8m^2+3m$, for fixed $m=n-k$, and they conjectured the sharp bound $n\le \left(\genfrac{}{}{0px}{}{m+2}{2}\right)$. Tuza proved the best known bound, $n\leq \frac{3}{4}m^2+m+1$,usingthemachineryof$\tau$-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 $n\le m^2+6m+2$. 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.