Crossing patterns of semi-algebraic sets

  • Authors:

  • Noga Alon;János Pach;Rom Pinchasi;Radoš Radoičić;Micha Sharir

  • Affiliations:

  • School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel;City College, CUNY and Courant Institute of Mathematical Sciences, New York University, NY;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA;Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA;School of Computer Science, Tel Aviv University, Tel Aviv, Israel, and Courant Institute of Mathematical Sciences, New York University, NY

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2005

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Abstract

We prove that, for every family F of n semi-algebraic sets in Rd 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 nδ 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.