The intersection spectrum of 3-chromatic intersecting hypergraph

The intersection spectrum of 3-chromatic intersecting hypergraph

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Benny Sudakov, ETH Zurich

The intersection spectrum of 3-chromatic intersecting hypergraphs For a hypergraph H, define its intersection spectrum I(H) as the set of all intersection sizes |E \cap F| of distinct edges E,F in H. In their seminal paper from 1973 which introduced the local lemma, Erdos and Lovasz asked: how large must the intersection spectrum of a k-uniform 3-chromatic intersecting hypergraph be? They showed that such a hypergraph must have at least three intersection sizes, and conjectured that the size of the intersection spectrum tends to infinity with k. Despite the problem being reiterated several times over the years by Erdos and other researchers, the lower bound of three intersection sizes has remarkably withstood any improvement until now. In this talk, we discuss the proof of the Erdos-Lovasz conjecture in a strong form which shows that there are at least k^{1/2-o(1)} intersection sizes. Joint work with M. Bucic and S. Glock.