Related papers: A counterexample to Borsuk's conjecture

For a finite set of natural numbers $D$ consider a complex polynomial of the form $f(z) = \sum_{d \in D} c_d z^d$. Let $\rho_+(f)$ and $\rho_-(f)$ be the fractions of the unit circle that $f$ sends to the right($\operatorname{Re} f(z) > 0$)…

A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set [n]={1,...,n}, denoted by 2^{[n]}, is at most…

We consider the volume of the largest axis-parallel box in the $d$-dimensional torus that contains no point of a given point set $\mathcal{P}_n$ with $n$ elements. We prove that, for all natural numbers $d, n$ and every point set…

If a family $\mathcal{F}$ of $k$-element subsets of an $n$-element set is pairwise intersecting, $2k\leq n$ then $|\mathcal{F}|\leq {n-1\choose k-1}$ holds by the celebrated Erd\H{o}s-Ko-Rado theorem. But an intersecting family obviously…

The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\|f\|_{U^d(V)}$ if and only if it…

We present a simple construction of an acute set of size $2^{d-1}+1$ in $\mathbb{R}^d$ for any dimension $d$. That is, we explicitly give $2^{d-1}+1$ points in the $d$-dimensional Euclidean space with the property that any three points form…

We prove that it is NP-hard to dissect one simple orthogonal polygon into another using a given number of pieces, as is approximating the fewest pieces to within a factor of $1+1/1080-\varepsilon$.

Let $F \subset \mathbb{R}^{2}$, and let $\dim_{\mathrm{A}}$ stand for Assouad dimension. I prove that $\dim_{\mathrm{A}} \pi_{e}(F) \geq \min\{\dim_{\mathrm{A}} F,1\}$ for all $e \in S^{1}$ outside of a set of Hausdorff dimension zero. This…

We prove the following sparse approximation result for polytopes. Assume that $Q$ is a polytope in John's position. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q'$ satisfies $Q \subseteq - 2d^2 \, Q'$. As a consequence,…

In this paper we study $N_d(k)$ the smallest positive integer such that any nice measure $\mu$ in $\R^d$ can be partitioned in $N_d(k)$ parts of equal measure so that every hyperplane avoids at least $k$ of them. A theorem of Yao and Yao…

A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is…

A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $\beta$ is one of those roots, then within one unit of $\beta$ lies a root of the polynomial's derivative. If we define $r(\beta)$ to be the…

We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby…

A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two…

Let $k$ and $d$ be such that $k \ge d+2$. Consider two $k$-colourings of a $d$-degenerate graph $G$. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al.…

We prove a generalization of the classical Borsuk--Ulam Theorem under small perturbations (shaking) of the sphere. We show that for a generic perturbation of a continuous map $f : S^2 \to \mathbb{R}^2$, the number of points $x \in S^2$ such…

We prove and generalize a conjecture in arXiv:1610.0474(4) about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$, where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit…

A conjecture of Armstrong states that if $\gcd (a, b) = 1$, then the average size of an $(a, b)$-core partition is $(a - 1)(b - 1)(a + b + 1) / 24$. Recently, Stanley and Zanello used a recursive argument to verify this conjecture when $a =…

For an integer $m >1$, we denote by $P(m)$ the largest prime divisor of $m$. We prove that $\limsup_{n \rightarrow +\infty} P(n!+1)/n \geqslant 1+9\log 2>7.238$, which improves a result of Stewart. More generally, for any nonzero polynomial…

The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…