 
Summary: Crossing Patterns of Semialgebraic Sets
Noga Alon
J´anos Pach
Rom Pinchasi§
Rados Radoici´c¶
Micha Sharir
Abstract
We prove that, for every family F of n semialgebraic sets in Rd
of constant description
complexity, there exist a positive constant that depends on the maximum complexity of the
elements of F, and two subfamilies F1, F2 F with at least n elements each, such that either
every element of F1 intersects all elements of F2 or no element of F1 intersects any element of
F2. This implies the existence of another constant such that F has a subset F F with n
elements, so that either every pair of elements of F intersect each other or the elements of F are
pairwise disjoint. The same results hold when the intersection relation is replaced by any other
semialgebraic relation. We apply these results to settle several problems in discrete geometry
and in Ramsey theory.
1 Introduction
Complete bipartite interaction in graph theory and in geometry. Let V (G) and E(G)
denote the vertex set and the edge set of a graph G, respectively. Let H be a fixed graph on k
