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arXiv · 2010.00495

The intersection spectrum of 3-chromatic intersecting hypergraphs

Matija Bucić·Stefan Glock·Benny Sudakov
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Abstract

For a hypergraph $H$, define its intersection spectrum $I(H)$ as the set of all intersection sizes $|E\cap F|$ of distinct edges $E,F\in E(H)$. In their seminal paper from 1973 which introduced the local lemma, Erd\H{o}s and Lov\'asz 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 Erd\H{o}s and other researchers, the lower bound of three intersection sizes has remarkably withstood any improvement until now. In this paper, we prove the Erd\H{o}s-Lov\'asz conjecture in a strong form by showing that there are at least $k^{1/2-o(1)}$ intersection sizes. Our proof consists of a delicate interplay between Ramsey type arguments and a density increment approach.

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Matija Bucić, Stefan Glock, Benny Sudakov. 2020-10-01. The intersection spectrum of 3-chromatic intersecting hypergraphs. https://arxiv.org/abs/2010.00495

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