The intersection spectrum of 3-chromatic intersecting hypergraphs

TL;DR

This paper proves a longstanding conjecture by Erdős and Lovász, showing that 3-chromatic intersecting hypergraphs have a large intersection spectrum, with at least roughly the square root of the uniformity parameter k of distinct intersection sizes.

Contribution

The paper establishes a lower bound of approximately k^{1/2} for the number of intersection sizes in 3-chromatic intersecting hypergraphs, confirming the Erdős-Lovász conjecture in a strong form.

Findings

At least k^{1/2 - o(1)} intersection sizes exist.

The proof combines Ramsey theory and density increment methods.

The longstanding conjecture is now proven with a near-optimal bound.

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.