Jeff Kahn, Gil Kalai
Let $f(d)$ be the smallest number so that every set in $R^d$ of diameter 1 can be partitioned into $f(d)$ sets of diameter smaller than 1. Borsuk's conjecture was that $f(d)\! =\!d\!+\!1$. We prove that $f(d)\! \ge\! (1.2)^{\sqrt d}$ for large~$d$.
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In this work, the classical Borsuk conjecture is discussed, which states that any set of diameter 1 in the Euclidean space $ {\mathbb R}^d $ can be divided into $ d+1 $ parts of smaller diameter. During the last two decades, many counterexamples to the conjecture have been proposed in high dimensions. However, all of them are sets of diameter 1 that lie on spheres whose radii are close to the value $ {1}{\sqrt{2}} $. The main result of this paper is as follows: {\it for any $ r > {1}{2} $, there exists a $ d_0 $ such that for all $ d \ge d_0 $, a counterexample to Borsuk's conjecture can be fo…
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…
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, …
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. …
An AI Generated Counterexample to Borsuk Problem in Dimension 63
该研究由ChatGPT利用GPT 5.6 Sol构造出63维含321个点的集合，证明b(63)≥65，为Borsuk问题提供了AI生成的反例。
Borsuk's conjecture for two-distance sets and its equivalent formulation for graphs
本研究利用图可作为双距离集嵌入欧氏空间的性质，给出双距离集下博苏克猜想的图论等价表述，介绍其未解决的维度范围，还探讨了图构造反例的思路与向s-距离集的推广。
Extremal spherical polytopes and Borsuk's conjecture
本文通过数值梯度流生成反自极球面多胞形，研究其临界点与组合性质，并探讨与Borsuk猜想的联系。
It is presented the simplest known disproof of the Borsuk conjecture stating that if a bounded subset of n-dimensional Euclidean space contains more than n points, then the subset can be partitioned into n+1 nonempty parts of smaller diameter. The argument is due to N. Alon and is a remarkable application of combinatorics and algebra to geometry. This note is purely expository and is accessible for students.
Sub-25-dimensional counterexamples to Borsuk's conjecture in the Leech lattice?
本文研究利奇格中Borsuk猜想的低维反例，通过构造24维利奇格及叠层子格中的最小范数向量集，验证n≤21时可分成至多n+1个小直径部分，并给出22至24维分别需要25、29、34部分的计算证据。
In this paper we answer Larman's question on Borsuk's conjecture for two-distance sets. We find a two-distance set consisting of 416 points on the unit sphere in the dimension 65 which cannot be partitioned into 83 parts of smaller diameter. This also reduces the smallest dimension in which Borsuk's conjecture is known to be false. Other examples of two-distance sets with large Borsuk's numbers will be given.
In this article we encode Hadwiger's covering conjecture and Borsuk's partition conjecture into continuous functions defined on the spaces of convex bodies, propose a four-step program to approach them, and obtain some partial results.
Borsuk and Vázsonyi problems through Reuleaux polyhedra
本文利用Reuleaux多面体建立了三维空间中Borsuk数为4的临界集与Vázsonyi问题极小结构之间的等价关系，并由此完整刻画了所有Borsuk数为4的有限集合。
In 1933, Borsuk proposed the following problem: Can every bounded set in $\mathbb{E}^n$ be divided into $n+1$ subsets of smaller diameters? This problem has been studied by many authors, and a lot of partial results have been discovered. In particular, Kahn and Kalai's counterexamples surprised the mathematical community in 1993. Nevertheless, the problem is still far away from being completely resolved. This paper presents a broad review on related subjects and, based on a novel reformulation, introduces a computer proof program to deal with this well-known problem.
In 1933, Borsuk conjectured that any bounded d-dimensional set of nonzero diameter can be broken into d + 1 parts of smaller diameter. This conjecture was disproved for large enough d, though it is true for low dimensional cases. The paper provides an alternative proof for d = 2 case.
Borsuk conjectured that every n-dimensional bounded set of positive diameter can be partitioned into n+1 sets of smaller diameters. This conjecture was proved for n=2 by Borsuk, for n=3 first by Eggleston, and disproved for n > 297 by Hinrichs and Richer. It is not known if the conjecture holds for 3 < n < 298. The best upper bound for the number of subsets of smaller diameters a four-dimensional set can be partitioned into is nine. This estimate was given by Lassak in 1982. In this note we improve this estimate by one.
The aim of this paper is to demonstrate relations between Gromov-Hausdorff distance properties and the Borsuk Conjecture. The Borsuk number of a given bounded metric space $X$ is the infimum of cardinal numbers $n$ such that $X$ can be partitioned into $n$ smaller parts (in the sense of diameter). An exact formula for the Gromov-Hausdorff distance between bounded metric spaces is obtained under the assumptions that the diameter and the cardinality of one space is less than the diameter and the Borsuk number of the other one, respectively. Using Bacon equivalence results between Lusternik-Schni…
The purpose of this article is to demonstrate the connection between the properties of the Gromov--Hausdorff distance and the Borsuk conjecture. The Borsuk number of a given bounded metric space $X$ is the infimum of cardinal numbers $n$ such that $X$ can be partitioned into $n$ smaller parts (in the sense of diameter). An exact formula for the Gromov--Hausdorff distance between bounded metric spaces is obtained under the assumption that the diameter and cardinality of one space are less than the diameter and Borsuk number of another, respectively. Using the results of Bacon's equivalence betw…
In 1933, Borsuk made a conjecture that every $n$-dimensional bounded set can be divided into $n+1$ subsets of smaller diameter. Up to now, the problem is still open for $4\leq n\leq 63$. In this paper, we firstly discuss the Banach-Mazur distance between the $n$-dimensional cube and the $\ell_{p}$ ball $(1\leq p< 2)$, then we study the generalized Borsuk's partition problem in metric spaces and prove that all bounded sets $X$ in every four-dimensional $\ell_{p}$ space can be divided into $2^4$ subsets of smaller diameter.
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…
In 1933, K. Borsuk proposed the following problem: Can every bounded set in $\mathbb{E}^n$ be divided into $n+1$ subsets of smaller diameters? In 1965, V. G. Boltyanski and I. T. Gohberg made the following conjecture: Every bounded set in an $n$-dimensional metric space can be divided into $2^n$ subsets of smaller diameters. In this paper, we prove the following result: Every bounded set in an $n$-dimensional metric space can be divided into $2^{n}((n+1)\log (n+1)+(n+1)\log \log (n+1)+5n+5)$ subsets of smaller diameters.
As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a centrally symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side length two. We discuss several counterexamples to this conjecture and, on the positive side, we identify a family of lattice polytopes that fulfill the claimed inequalities. This family is related to the recently introduced class of $l$-reflexive polytopes.
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R^d}$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z^2} \times G_0$ for some finite abelian $G_0$. Our methods rely on encoding a certain class of "$p$-adically structured functions" in terms of certain functional equations.
Generalized Borsuk Graphs
本文定义了紧致度量空间上的$G$-Borsuk图，证明其小$ε$下的色数由拓扑$G$-覆盖数决定，并利用Hom-复形与维数递归公式分别给出色数下界与上界，同时探究了随机诱导子图的色数阈值。
Paper withdrawn by the author.
A Counterexample to Viterbo's Conjecture
本文通过一个反例证明了维特博猜想不成立，揭示了辛容量在凸域上的不一致性。
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