Abstract
We continue a sequence of recent works studying Ramsey functions for semialgebraic predicates in R-d. A k-ary semialgebraic predicate Phi(x(1), ..., x(k)) on R-d is a Boolean combination of polynomial equations and inequalities in the kd coordinates of k points x(1), ..., x(k) is an element of R-d.
A sequence P = (p(1), ..., p(n)) of points in R-d is called Phi-homogeneous if either Phi(p(i1), ..., p(ik)) holds for all choices 1 {= i(1) < ... < i(k) {= n, or it holds for no such choice. The Ramsey function R-Phi(n) is the smallest N such that every point sequence of length N contains a Phi-homogeneous subsequence of length n.
Conlon et al. [Trans. Amer.
Math. Soc., 366 (2013), pp. 5043-5065] constructed the first examples of semialgebraic predicates with the Ramsey function bounded from below by a tower function of arbitrary height: for every k }= 4, they exhibit a k-ary Phi in dimension 2(k-4) with R-Phi bounded below by a tower of height k - 1.
We reduce the dimension in their construction, obtaining a k-ary semialgebraic predicate Phi on Rk-3 with R-Phi bounded below by a tower of height k - 1. We also provide a natural geometric Ramsey-type theorem with a large Ramsey function.
We call a point sequence P in R-d order-type homogeneous if all (d + 1)-tuples in P have the same orientation. Every sufficiently long point sequence in general position in R-d contains an order-type homogeneous subsequence of length n, and the corresponding Ramsey function has recently been studied in several papers.
Together with a recent work of Barany, Matousek, and Por [Curves in R-d Intersecting Every Hyperplane at Most d + 1 Times, preprint, arXiv:1309.1147; extended abstract in Proceedings of the 30th Annual Symposium on Computational Geometry, 2014], our results imply a tower function of Omega(n) of height d as a lower bound, matching an upper bound by Suk up to the constant in front of n.