相关论文: Monochromatic sums equal to products near zero

If E is a linear homogenous equation and c a natural then the Rado number $R_c(E)$ is the least N so that any c-coloring of the positive integers from 1 to N contains a monochromatic solution. Rado characterized for which E R_c(E) always…

Let $[a,b]$ denote the integers between $a$ and $b$ inclusive and, for a finite subset $X \subseteq \mathbb{Z}$, let the diameter of $X$ be equal to $\max(X)-\min(X)$. We write $X<_p\,Y$ provided $\max(X)<\min(Y)$. For a positive integer…

Given integers $m\le c$ and an exact $c$-coloring of the edges of a complete countably infinite graph (i.e. a coloring that uses exactly $c$ colors), must there be an infinite subgraph that is exactly $m$-colored? Using the Infinite Ramsey…

The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any $n$-uniform simple (i.e. every two distinct edges share at most one vertex) hypergraph $H$ with maximum edge…

We prove a known 2-coloring of the integers $[N] := \{1,2,3,\ldots,N\}$ minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers…

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength}…

We address the question of the "partition regularity" of the Pythagorean equation a^2+b^2=c^2; in particular, can the natural numbers be assigned a 2-coloring, so that no Pythagorean triple (i.e., a solution to the equation) is…

We improve a result of Solymosi on sum-products in R, namely, we prove that max{|A+A|,|AA|}\gg |A|^{4/3+c}, where c>0 is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved…

Recent results of Hindman, Leader and Strauss and of Fern\'andez-Bret\'on and Rinot showed that natural versions of Hindman's Theorem fail {\em for all} uncontable cardinals. On the other hand, Komj\'ath proved a result in the positive…

Motivated by questions asked by Erdos, we prove that any set $A\subset{\mathbb N}$ with positive upper density contains, for any $k\in{\mathbb N}$, a sumset $B_1+\cdots+B_k$, where $B_1,\dots,B_k\subset{\mathbb N}$ are infinite. Our proof…

We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\{x,y,x+ay\}$ triples for $a \geq 1$. We give a new proof of the original case of $a=1$. We show that the minimum…

Let $t\geq 1$, let $A$ and $B$ be finite, nonempty subsets of an abelian group $G$, and let $A\pp{i} B$ denote all the elements $c$ with at least $i$ representations of the form $c=a+b$, with $a\in A$ and $b\in B$. For $|A|, |B|\geq t$, we…

A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, $\{(a_i,b_i,c_i)\}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by…

We study a generalization of a famous result of Goodman and establish that asymptotically at least a $1/256$ fraction of all triangles needs to be monochromatic in any four-coloring of the edges of a complete graph. We also show that any…

Let $S \subset \mathbb{Z}^{d}$ be a finitely generated subsemigroup. Let $E$ be a product system over $S$. We show that there exists an infinite dimensional separable Hilbert space $\mathcal{H}$ and a semigroup $\alpha:=\{\alpha_x\}_{x \in…

We show that for any finite coloring of the group $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ and for any positive integer $k$, there always exists a monochromatic regular $k$-gon in $\mathbb{Z}_2 *\mathbb{Z}_2 *\mathbb{Z}_2$ with respect to…

Let $r(k,A,n)$ denote the number of representations of $n$ as a sum of $k$ elements of a set $A \subseteq \mathbb{N}$. In 2002, Dombi conjectured that if $A$ is co-infinite, then the sequence $(r(k,A,n))_{n \geq 0}$ cannot be strictly…

The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and…

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after…