In this paper we develop a concrete way to construct bodies of constant width in dimension three. They are constructed from special embeddings of selfdual graphs.
We study nested partitions of R d obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing the parts among k sets. This generalises classical necklace splitting results and their more recent high-dimensional versions. With similar methods we show that in the plane, for any t measures there is a path formed only by horizontal and vertical segments using at most t−1 turns that splits them by half simultaneously, and optimal mass-partitioning results for chessboard colourings of R d using hyperplanes with fixed directions.
We continue a sequence of recent works studying Ramsey functions for semialgebraic predicates in R d . A k-ary semialgebraic predicate Φ(x 1 , . . . , x k ) on R d is a Boolean combination of polynomial equations and inequalities in the kd coordinates of k points. . , p i k ) holds for all choices 1 ≤ i 1 < · · · < i k ≤ n, or it holds for no such choice. The Ramsey function R Φ (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 2 k−4 with R Φ bounded below by a tower of height k − 1. We reduce the dimension in their construction, obtaining a k-ary semialgebraic predicate Φ on R k−3 with R Φ 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 R d order-type homogeneous if all (d + 1)-tuples in P have the same orientation. Every sufficiently long point sequence in general position in R d 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.
In this paper we give an asymptotically tight bound for the tolerated Tverberg Theorem when the dimension and the size of the partition are fixed. To achieve this, we study certain partitions of order-type homogeneous sets and use a generalization of the Erdős-Szekeres theorem.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.