Borsuk's conjecture states that any bounded set in R^n can be partitioned into n+1 sets of smaller diameter. It is known to be false for all n bigger or equal to 323.
Here we show that Borsuk's conjecture fails in dimensions 321 and 322. (This result has been independently discovered by Hinrichs and Richter.)
A Chain-Level Borsuk--Ulam Obstruction Proof of Norine's Antipodal-Coloring Conjecture
Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极值组合与概率研究组）
In 1933 Karol Borsuk asked whether each bounded set in the n-dimensional Euclidean space can be divided into n+1 parts of smaller diameter. The diameter of a set is defined as the supremum (least upper bound) of the distances of contained points. Implicitly, the whole set is assumed to contain at least two points. The hypothesis that the answer to that question is positive became famous under the name Borsuk's conjecture. Beginning with Jeff Kahn and Gil Kalai, from 1993 to 2003 several authors have proved that in certain (almost all) high dimensions such a division is not generally possible. …
In the concluding remarks of their 1993 published and now famous paper, Jeff Kahn and Gil Kalai wrote in particular: "Our construction shows that Borsuk's conjecture is false for d = 1,325 and for every d > 2,014." But, as Bernulf Weiszbach remarked in his paper from 2000, a simple (few steps for an upper-class pocket calculator) computation indicates that that claim is not true for d=1325. William Kretschmer (Univ. of Texas) sent me his writeup on that paper by Kahn and Kalai, in particular pointing out that the derivation of the formula used in that computation disregarded a certain aspect, …
We give a short survey of problems and results on (1) diameter graphs and hypergraphs, and (2) geometric Ramsey theory. We also make some modest contributions to both areas. Extending a well known theorem of Kahn and Kalai which disproved Borsuk's conjecture, we show that for any integer $r\ge 2$, there exist $\varepsilon=\varepsilon(r)>0$ and $d_0=d_0(r)$ with the following property. For every $d\ge d_0$, there is a finite point set $P\subset\mathbb{R}^d$ of diameter $1$ such that no matter how we color the elements of $P$ with fewer than $(1+\varepsilon)^{\sqrt{d}}$ colors, we can always fin…
In 2013 Andriy V. Bondarenko showed how to construct a two-distance counterexample to Borsuk's conjecture from any strongly regular graph whose vertex set is not the union of at most $f+1$ cliques (sets of pairwise adjacent vertices) where $f$ is the multiplicity of the second-largest eigenvalue of its adjacency matrix. He applied that construction to those two graphs that he had been able to prove to fulfill the condition: From the $G_2(4)$ graph (on 416 vertices) he got a 65-dimensional two-distance counterexample. From the $Fi_{23}$ graph (on 31671 vertices) he got a 782-dimensional one and…
Bárány's "topological Tverberg conjecture" from 1976 states that any continuous map of an $N$-simplex $Δ_N$ to $\mathbb{R}^d$, for $N\ge(d+1)(r-1)$, maps points from $r$ disjoint faces in $Δ_N$ to the same point in $\mathbb{R}^d$. The proof of this result for the case when $r$ is a prime, as well as some colored version of the same result, using the results of Borsuk-Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matoušek's 2003 book "Using the Borsuk-Ulam theorem." In this paper we show how advanced equivariant topology meth…
Hedetniemi conjectured in 1966 that $χ(G \times H) = \min\{χ(G), χ(H)\}$ for all graphs $G$ and $H$. Here $G\times H$ is the graph with vertex set $ V(G)\times V(H)$ defined by putting $(x,y)$ and $(x',y')$ adjacent if and only if $xx'\in E(G)$ and $yy'\in E(H)$. This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let $p$ be the minimum number of vertices in a graph of odd girth $7$ and fractional chromatic number greater than $3+4/(p-1)$. Shitov's proof shows that Hedetniemi's conjecture fails for some graphs with chromatic number ab…
We describe computer programs accompanying our paper and show that running them suffices to verify Fuglede's conjecture in $\mathbb{Z}_2^5$ and $\mathbb{Z}_2^6$.
Neither simpliciality nor mutation connectivity: conjectures of Las Vergnas and Cordovil-Las Vergnas fail
University of Chicago（芝加哥大学）；Departament de Matemàtiques i Informàtica, Universitat de Barcelona（巴塞罗那大学数学与计算机系）；Centre de Recerca Matemàtica (CRM)（数学研究中心）
Although the Unimodality Conjecture holds for some certain classes of cubical polytopes (e.g. cubes, capped cubical polytopes, neighborly cubical polytopes), it fails for cubical polytopes in general. A 12-dimensional cubical polytope with non-unimodal face vector is constructed by using capping operations over a neighborly cubical polytope with 2 to the power 131 vertices. For cubical polytopes, the Unimodality Conjecture is proved for dimensions less than 11. The first one-third of the face vector of a cubical polytope is increasing and its last one-third is decreasing in any dimension.
Common neighbour conjectures for Saxl graphs fail at every base size
King’s College London（伦敦国王学院）；Heilbronn Institute for Mathematical Research（希尔伯恩数学研究所）；Warwick Mathematics Institute, University of Warwick（华威大学华威数学研究所）
Sellami and Sirvent conjectured that the balanced pair algorithm fails for the following pair of Pisot substitutions: \[ φ_0: \begin{array}{l} a \mapsto abc b \mapsto a c \mapsto ac \end{array} \quad \text{ and } \quad φ_1: \begin{array}{l} a \mapsto cba b \mapsto a c \mapsto ca \end{array}. \] The conjecture stated the balanced pair algorithm fails because there is no initial balanced pair. In the present note we prove this conjecture using a method based on simultaneous coding of the pair of the fixed points of the morphisms.
