Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, June 18, 2013, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Marek Elias (Charles University)

Some Ramsey questions in discrete geometry

In 1935 Erdos and Szekeres proved that in a set of n^2 points in a plane n of them form a monotonous subset. They also proved that in a set of 2^n points there is a convex or a concave subset of size n. A semialgebraic predicate of arity k in dimension d is a predicate on k points from R^d which is described by a logical formula whose atoms are polynomial inequalities. F(x,y) = (x_1 - y_1 >= 0) && (x2 - y2 >= 0) can be one such example. In Ramsey theory we are looking for homogeneous subsets with respect to F i. e. having all k-tuples satisfying F or all k-tuples not satisfying F. We will present a result of a paper with Jiri Matousek on generalization of the theorems of Erdos and Szekeres as well as more recent results on Ramsey numbers for semialgebraic predicates of Bukh and Matousek, and Conlon et al, and a new paper on order-types by Suk.

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