@inbook{48386f8f9a2b46a39d70dd113c272756,

title = "The Intersection Spectrum of 3-Chromatic Intersecting Hypergraphs",

abstract = "For a hypergraph H, define its intersection spectrum I(H) as the set of all intersection sizes | E∩ F| of distinct edges E, F∈ E(H). In their seminal paper from 1973 which introduced the local lemma, Erd{\H o}s and Lov{\'a}sz 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{\'a}sz conjecture in a strong form by showing that there are at least k1 / 2 - o ( 1 ) intersection sizes. In this extended abstract we sketch a simpler argument which gives slightly weaker bound of k1 / 3 - o ( 1 ). Our proof consists of a delicate interplay between Ramsey type arguments and a density increment approach.",

keywords = "Intersecting hypergraphs, Intersection spectrum, Property B",

author = "Matija Buci{\'c} and Stefan Glock and Benny Sudakov",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.",

year = "2021",

doi = "10.1007/978-3-030-83823-2_22",

language = "English (US)",

series = "Trends in Mathematics",

publisher = "Springer Science and Business Media Deutschland GmbH",

pages = "136--141",

booktitle = "Trends in Mathematics",

address = "Germany",

}