Non-representability of finite projective planes by convex sets

We prove that there is no d such that all finite projective planes can be represented by convex sets in R, answering a question of Alon, Kalai, Matoušek, and Meshulam. Here, if P is a projective plane with lines l1, . . . , ln, a representation of P by convex sets in R is a collection of convex sets C1, . . . , Cn ⊆ R d such that Ci1 , Ci2 . . . , Cik have a common point if and only if the corresponding lines li1 , . . . , lik have a common point in P. The proof combines a positive-fraction selection lemma of Pach with a result of Alon on “expansion” of finite projective planes. As a corollary, we show that for every d there are 2-collapsible simplicial complexes that are not d-representable, strengthening a result of Matoušek and the author.