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Abstract
Given a finite point set , a -ary semi-algebraic relation on is the set of -tuples of points in , which is determined by a finite number of polynomial equations and inequalities in real variables. The description complexity of such a relation is at most if the number of polynomials and their degrees are all bounded by . The Ramsey number is the minimum such that any -element point set in equipped with a -ary semi-algebraic relation , such that has complexity at most , contains members such that every -tuple induced by them is in , or members such that every -tuple induced by them is not in . We give a new upper bound for for and fixed. In particular, we show that for fixed integers , …
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