The Borsuk-Ulam-property, Tucker-property and constructive proofs in combinatorics

This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tucker-property of a finite group $G$ is introduced and its relation to the topological Borsuk-Ulam-property is discussed. Applications of the Tucker-property in combinatorics are demonstrated.Comment: 12 pages, 0 figure

Balanced Islands in Two Colored Point Sets in the Plane

Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $\alpha \in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil \alpha r\rceil$ red points and exactly $\lceil \alpha b \rceil$ blue points of $S$; a convex set containing exactly $\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an $O(n^4)$ time algorithm and an $O(n^2\log n)$ time algorithm in the second case. Finally, if $\lceil \alpha r\rceil+\lceil \alpha b\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$

A Constructive Proof of Ky Fan\u27s Generalization of Tucker\u27s Lemma

We present a proof of Ky Fan\u27s combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan\u27s lemma to allow for triangulations of Sn that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker\u27s lemma that holds for a more general class of triangulations than the usual version

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the $g$-conjecture for spheres (as a talk and as a "Q\&A" evening session) (2) Federico Ardila gave an overview on "The geometry of matroids", including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz

Approximation algorithms for low-distortion embeddings into low-dimensional spaces

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 33-35).We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the two-dimensional plane. We give an O([square root] n)-approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. For the same problem, we give an exact algorithm, with running-time exponential in the distortion. We complement these results by showing that the problem is NP-hard to [alpha]-approximate, for some constant [alpha] > 1. For the two-dimensional case, we show a O([square root] n) upper bound for the distortion required to embed an n-point subset of the two-dimensional sphere, into the plane. We prove that this bound is asymptotically tight, by exhibiting n-point subsets such that any embedding into the plane has distortion [omega]([square root] n). These techniques yield a O(1)-approximation algorithm for the problem of embedding an n-point subset of the sphere into the plane.by Anastasios Sidiropoulos.S.M

Multilabeled versions of Sperner's and Fan's lemmas and applications

We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner's lemma. A notable application of this technique yields a cake-cutting theorem where the number of players and the number of pieces can be independently chosen. We also prove multilabeled versions of Fan's lemma, a combinatorial analogue of the Borsuk-Ulam theorem, and exhibit applications to fair division and graph coloring.Comment: 21 pages, 2 figure