The intersection spectrum of 3-chromatic intersecting hypergraphs

For a hypergraph H$H$, define its intersection spectrum I(H)$I(H)$ as the set of all intersection sizes |E boolean AND F|$|E\cap F|$ of distinct edges E,F is an element of E(H)$E,F\in E(H)$. In their seminal paper from 1973 which introduced the local lemma, Erdos and Lovasz asked: how large must the intersection spectrum of a k$k$-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$k$. 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 paper, we prove the Erdos-Lovasz conjecture in a strong form by showing that there are at least k1/2-o(1)$k{1/2-o(1)}$ intersection sizes. Our proof consists of a delicate interplay between Ramsey-type arguments and a density increment approach.
Proceedings of the London Mathematical Society, 124 (5)
ISSN:0024-6115
ISSN:1460-244X