Crossing patterns of semi-algebraic sets

We prove that, for every family F of n semi-algebraic sets in R of constant description complexity, there exist a positive constant ε that depends on the maximum complexity of the elements of F , and two subfamilies F1,F2 ⊆ F with at least εn 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 δ 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 semi-algebraic relation. We apply these results to settle several problems in discrete geometry and in Ramsey theory.