Chromatic numbers of hypergraphs encode nontrivial information about the intersection pattern of finite sets. Here Kneser hypergraphs give information about the intersections among all k-subsets of an n-set. Since Lovasz' groundbreaking proof of the Kneser conjecture these problems have been approached with methods from geometry and topology. In this talk we will establish a correspondence between intersection patterns of finite sets and of convex sets in Euclidean space. The latter can be understood via equivariant topology and we thus establish new lower bounds for chromatic numbers.