The vector graph and the chromatic number of the plane, or how NOT to prove that χ(E2)>4\chi(\mathbb{E}^2)>4

The chromatic number $\chi\left(\mathcal{E^2}\right)$ of the plane is known to be some integer between 4 and 7, inclusive. We prove a limiting result that says, roughly, that one cannot increase the lower bound on $\chi\left(\mathcal{E^2}\right)$ by pasting Moser Spindles together, even countably many.Comment: To appear in Australasian Journal of Combinatoric

Isometry Group of Gromov--Hausdorff Space

The present paper is devoted to investigation of the isometry group of the Gromov-Hausdorff space, i.e., the metric space of compact metric spaces considered up to an isometry and endowed with the Gromov-Hausdorff metric. The main goal is to present a proof of the following theorem by George Lowther (2015): The isometry group of the Gromov-Hausdorff space is trivial. Unfortunately, the author himself has not publish an accurate text for 2 years passed from the publication of draft (that is full of excellent ideas mixed with unproved and wrong statements) in the https://mathoverflow.net/ blog (see the exact reference in he bibliography).Comment: 28 pages, 4 figures, 13 bib item

Some old and new problems in combinatorial geometry I: Around Borsuk's problem

Borsuk asked in 1933 if every set of diameter 1 in $R^d$ can be covered by $d+1$ sets of smaller diameter. In 1993, a negative solution, based on a theorem by Frankl and Wilson, was given by Kahn and Kalai. In this paper I will present questions related to Borsuk's problem.Comment: This is a draft of a chapter for "Surveys in Combinatorics 2015," edited by Artur Czumaj, Angelos Georgakopoulos, Daniel Kral, Vadim Lozin, and Oleg Pikhurko. The final published version shall be available for purchase from Cambridge University Pres

Unit distance graphs with ambiguous chromatic number

First Laszlo Szekely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with vertex set R^n, and hence may be seen as further evidence that the chromatic number of the plane might depend on set theory.Comment: 7 page

Invertibility via distance for non-centered random matrices with continuous distributions

Let $A$ be an $n\times n$ random matrix with independent rows $R_1(A),\dots,R_n(A)$, and assume that for any $i\leq n$ and any three-dimensional linear subspace $F\subset {\mathbb R}^n$ the orthogonal projection of $R_i(A)$ onto $F$ has distribution density $\rho(x):F\to{\mathbb R}_+$ satisfying $\rho(x)\leq C_1/\max(1,\|x\|_2^{2000})$ ($x\in F$) for some constant $C_1>0$. We show that for any fixed $n\times n$ real matrix $M$ we have $${\mathbb P}\{s_{\min}(A+M)\leq t n^{-1/2}\}\leq C'\, t,\quad\quad t>0,$$ where $C'>0$ is a universal constant. In particular, the above result holds if the rows of $A$ are independent centered log-concave random vectors with identity covariance matrices. Our method is free from any use of covering arguments, and is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar-Spielman-Teng for non-centered Gaussian matrices.Comment: revised versio

On a generalization of the Hadwiger-Nelson problem

For a field $F$ and a quadratic form $Q$ defined on an $n$-dimensional vector space $V$ over $F$, let $\mathrm{QG}_Q$, called the quadratic graph associated to $Q$, be the graph with the vertex set $V$ where vertices $u,w \in V$ form an edge if and only if $Q(v-w)=1$. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In the present paper, we will prove that for a local field $F$ of characteristic zero, the Borel chromatic number of $\mathrm{QG}_Q$ is infinite if and only if $Q$ represents zero non-trivially over $F$. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009.Comment: This is the final version. Accepted in Israel Journal of Mathematic

MACRO: A Meta-Algorithm for Conditional Risk Minimization

We study conditional risk minimization (CRM), i.e. the problem of learning a hypothesis of minimal risk for prediction at the next step of sequentially arriving dependent data. Despite it being a fundamental problem, successful learning in the CRM sense has so far only been demonstrated using theoretical algorithms that cannot be used for real problems as they would require storing all incoming data. In this work, we introduce MACRO, a meta-algorithm for CRM that does not suffer from this shortcoming, but nevertheless offers learning guarantees. Instead of storing all data it maintains and iteratively updates a set of learning subroutines. With suitable approximations, MACRO applied to real data, yielding improved prediction performance compared to traditional non-conditional learning

Measurable Chromatic Number of Spheres

We examine the measurable chromatic number of distance colorings on the surface of 2-dimensional spheres of varying radii, showing in particular that similar arguments to those used to raise lower bounds in the plane work for all but a countable set of radii. Furthermore, we show that measurable chromatic number as a function of the radius, or more generally the curvature, is not monotonic.Comment: 9 pages, 1 figur

Measurable sets with excluded distances

For a set of distances D={d_1,...,d_k} a set A is called D-avoiding if no pair of points of A is at distance d_i for some i. We show that the density of A is exponentially small in k provided the ratios d_1/d_2, d_2/d_3, ..., d_{k-1}/d_k are all small enough. This resolves a question of Szekely, and generalizes a theorem of Furstenberg-Katznelson-Weiss, Falconer-Marstrand, and Bourgain. Several more results on D-avoiding sets are presented.Comment: 23 pages, 3 figures, typos and small errors fixe

Better bounds for planar sets avoiding unit distances

A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ ($n\ge 2$) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than $1$ and points from distinct blocks lie farther than $1$ unit of distance apart from each other) has density strictly less than $1/2^n$. For the special case of sets with block structure this proves a conjecture of Erd\H{o}s asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$.Comment: 16 pages, 1 figure. Contains a Sage script called dstverify.sage, to verify the application of Theorem 3.3. Download the article source to get the scrip