Semi-algebraic Ramsey numbers

Abstract

Given a finite point set P⊂Rd, a k-ary semi-algebraic relation E on P is a set of k-tuples of points in P 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. The Ramsey number Rkd,t(s,n) is the minimum N such that any N-element point set P in Rd equipped with a k-ary semi-algebraic relation E of complexity at most t contains s members such that every k-tuple induced by them is in E or n members such that every k-tuple induced by them is not in E.We give a new upper bound for Rkd,t(s,n) for k≥3 and s fixed. In particular, we show that for fixed integers d,t,sR3d,t(s,n)≤2no(1), establishing a subexponential upper bound on R3d,t(s,n). This improves the previous bound of 2nC1 due to Conlon, Fox, Pach, Sudakov, and Suk where C1 depends on d and t, and improves upon the trivial bound of 2nC2 which can be obtained by applying classical Ramsey numbers where C2 depends on s. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in Rd. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results.