Semi-algebraic Ramsey numbers

8 years 14 days ago

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Given a ﬁnite point set P ⊂ Rd , a k-ary semi-algebraic relation E on P is the set of k-tuples of points in P, 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. The Ramsey number Rd,t k (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 such that E has 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 Rd,t k (s, n) for k ≥ 3 and s ﬁxed. In particular, we show that for ﬁxed integers d, t, s Rd,t 3 (s, n) ≤ 2no(1) , establishing a subexponential upper bound on Rd,t 3 (s, n). This improves the previous bound of 2nC due to Conlon, Fox, Pach, Sudakov, and Suk, where C is a very large constant depending on ...

Andrew Suk

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