The Szemer\'edi-Petruska conjecture for a few small values

TL;DR

This paper verifies the Szemerédi-Petruska conjecture for small values of m in 3-uniform hypergraphs, establishing the sharp bound on the order of the hypergraph under specific clique intersection conditions.

Contribution

The paper confirms the Szemerédi-Petruska conjecture for m=2, 3, and 4, providing new proofs for these small cases.

Findings

Conjecture holds for m=2, 3, 4

Established sharp bounds for these cases

Advances understanding of hypergraph clique structures

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, Szemer\'edi and Petruska conjectured the sharp bound $n\leq {m+2\choose 2}$. In this note the conjecture is verified for m=2,3 and 4.