Ramsey-type results for semi-algebraic relations

A k-ary semi-algebraic relation E on R^d is a subset of R^{kd}, the set of k-tuples of points in R^d, which is determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a relation is at most t if the number of polynomials and their degrees are all bounded by t. A subset A of R^d is called homogeneous if all or none of the k-tuples from A satisfy E. A large number of geometric Ramsey-type problems and results can be formulated as questions about finding large homogeneous subsets of sets in R^d equipped with semi-algebraic relations. In this paper we study Ramsey numbers for k-ary semi-algebraic relations of bounded complexity and give matching upper and lower bounds, showing that they grow as a tower of height k-1. This improves on a direct application of Ramsey's theorem by one exponential and extends a result of Alon, Pach, Pinchasi, Radoi\v{c}i'c, and Sharir, who proved this for k=2. We apply our results to obtain new estimates for some geometric Ramsey-type problems relating to order types and one-sided sets of hyperplanes. We also study the off-diagonal case, achieving some partial results.