Edgardo Roldán-Pensado

A characteristic property of the Euclidean disc

AbstractIn this paper the following is proved: Let K ⊂

Lower Bounds on Geometric Ramsey Functions

\mathbb{E}^2

Measure partitions using hyperplanes with fixed directions

be a smooth strictly convex body, and let L ⊂

Infinite numerical range is convex

\mathbb{E}^2

Cutting convex curves

be a line. Assume that for every point x ∈ L/K the two tangent segments from x to K have the same length, and the line joining the two contact points passes through a fixed point in the plane. Then K is an Euclidean disc.

The colored Hadwiger transversal theorem in źd

We continue a sequence of recent works studying Ramsey functions for semialgebraic predicates in Rd. A k-ary semialgebraic predicate Φ(x1, …, xk) on Rd is a Boolean combination of polynomial equations and inequalities in the kd coordinates of k points x1, …, xk ∈ Rd. A sequence P = (p1, …, pn) of points in Rd is called Φ-homogeneous if either Φ(pi1, …,pik) holds for all choices 1 ≤ i1 < … < ik ≤ n, or it holds for no such choice. The Ramsey function RΦ (n) is the smallest N such that every point sequence of length N contains a Φ-homogeneous subsequence of length n. Conlon, Fox, Pach, Sudakov, and Suk 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 Φ in dimension 2k−4 with RΦ bounded below by a tower of height k − 1. We reduce the dimension in their construction, obtaining a k-ary semialgebraic predicate Φ on Rk−3 with RΦ 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 Rd order-type homogeneous if all (d + 1)-tuples in P have the same orientation. Every sufficiently long point sequence in general position in Rd 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 Bárány, Matoušek, and Pór, our results imply a tower function of Ω(n) of height d as a lower bound, matching an upper bound by Suk up to the constant in front of n.