Semi-algebraic Ramsey numbers

Given a finite point set P⊂RdP⊂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 RdRd 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≥3k≥3 and s   fixed. In particular, we show that for fixed integers d,t,sd,t,sR3d,t(s,n)≤2no(1), establishing a subexponential upper bound on R3d,t(s,n). This improves the previous bound of 2nC12nC1 due to Conlon, Fox, Pach, Sudakov, and Suk where C1C1 depends on d and t  , and improves upon the trivial bound of 2nC22nC2 which can be obtained by applying classical Ramsey numbers where C2C2 depends on s  . As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in RdRd. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results.