  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.
