English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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 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.

Lower bounds on geometric Ramsey functions