Related papers: Tverberg's theorem, a new proof

We first establish a general random Sperner lemma by presenting a completely new approach for the theory of $L^{0}$-simplicial subdivisions of $L^{0}$-simplexes. Based on this, we are able to achieve a new complete proof of the random…

Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

Erd\H{o}s-Ginzburg-Ziv theorem says that if there are 2n-1 number is given, then there are n numbers such that their sum is divided by n. We will connect this theorem with the Ramsey theoretic large sets and will prove an infinitary version…

In this paper we consider permutations of sequences of partitions, obtaining a result which parallels von Neumann's theorem on permutations of dense sequences and uniformly distributed sequences of points.

We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1,…

In Sternfeld's work on Kolmogorov's Superposition Theorem appeared the combinatorial-geometric notion of a basic set and a certain kind of arrays. A subset $X \subset \mathbb R^n$ is basic if any continuous function $X\to \mathbb R$ could…

We generalize the ham sandwich theorem to $d+1$ measures in $\mathbb{R}^d$ as follows. Let $\mu_1,\mu_2, \dots, \mu_{d+1}$ be absolutely continuous finite Borel measures on $\mathbb{R}^d$. Let $\omega_i=\mu_i(\mathbb{R}^d)$ for $i\in…

This document presents an alternative proof of Sylvester's theorem stating that "the product of $n$ consecutive numbers strictly greater than $n$ is divisible by a prime strictly greater than $n$". In addition, the paper proposes stronger…

We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1,2,...,n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the…

The main purpose of this paper is to present a new and more uniform model-theoretic/combinatorial proof of the theorem ([5]): The randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$ is a (complete) first-order continuous…

Using the Hilbert-Schmidt theorem, we reformulate the R-matrix theory in terms of a uniformly and absolutely convergent expansion. Term by term differentiation is possible with this expansion in the neighborhood of the surface. Methods for…

We prove some analogues of the central point theorem and Tverberg's theorem, where instead of points, we consider hyperplanes or affine flats of given dimension.

A \textsl{Sperner $k$-partition system} on a set $X$ is a set of partitions of $X$ into $k$ classes such that the classes of the partitions form a Sperner set system (so no class from a partition is a subset of a class from another…

Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of…

In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove…

We prove the following uniform version of a theorem by Lindstr\"om: Let $\mbox{$\cal F$}:=\{F_i:~ i\in I\}$ be a $k$-uniform set family of $[n]$, where $k\geq 1$. If $|\mbox{$\cal F$}|\geq n+1$, then there exist two disjoint subsets $I_1$…

We give a sharpened form of Siegel Lemma's w. r. t. the maximum norm. This implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erd\"os-Moser problem). The main tools are Minkowski's theorem on…

Partition theory abounds with bijections between different types of partitions. One of the most famous partition bijections maps each self-conjugate partition of a positive integer $n$ to a partition of $n$ into distinct odd parts, and vice…

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that…