An asymptotic resolution of a conjecture of Szemerédi and Petruska
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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≤ 8m^2+3m, for fixed m=n-k, and they conjectured the sharp bound n ≤m+2 2. 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: arXiv:2204.02859). The best known bound, n≤3/4m^2+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+2 2 + O(m^5/3), resolving the conjecture asymptotically.
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