Ramsey-type results for semi-algebraic relations

Understand

A k-ary semi-algebraic relation E on R^d is a subset of R^{kd}, the set of k-tuples of points in R^d, 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.
• A subset A of R^d is called homogeneous if all or none of the k-tuples from A satisfy E.
• A large number of geometric Ramsey-type problems and results can be formulated as questions about finding large homogeneous subsets of sets in R^d equipped with semi-algebraic relations.