Related papers: A Multidimensional Rado Theorem

For a positive integer $m$ and a real number $c$, let $R = R(m,c,2)$ denote the discrete 2-color Rado number for the equation $x_1 + x_2 + \dots + x_m + c = 2x_0$. In other words, $R$ is the smallest integer such that for any coloring of…

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards…

In this article, we investigate homogeneous versions of certain nonlinear Ramsey-theoretic results, with three significant applications. As the first application, we prove that for every finite coloring of $\mathbb{Z}^+$, there exist an…

An equation is called graph-regular if it always has monochromatic solutions under edge-colorings of the complete graph on the naturals. We present two Rado-like conditions which are respectively necessary and sufficient for an equation to…

We show that for non-zero integers $a$ and $b$ there is a natural number $N < \exp(r^{2+o_{a,b;r\rightarrow \infty}(1)})$ such that in any $r$-colouring of $\{1,\dots,N\}$ there are $x,y,z$, all in the same colour class, such that…

The purpose of this paper is to present some multidimensional fixed-point theorems and their applications. For this, we provide a multidimensional fixed point theorem and then using this theorem we prove the existence and uniqueness of a…

In 1982, Beutelspacher and Brestovansky determined the 2-color Rado number of the equation $$x_1+x_2+\cdots +x_{m-1}=x_m$$ for all $m\geq 3.$ Here we extend their result by determining the 2-color Rado number of the equation…

Ramsey's theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey's theorem for pairs due to Erdos and Rado, which states that every…

Van der Waerden's (VDW) colouring theorem in combinatoric number theory [1] has scope for physical applications.The solution of the two colour case has enabled the construction of an explicit mapping of an infinite, one dimensional…

Hadwiger's transversal theorem gives necessary and sufficient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful…

A well-known consequence of Schur's theorem is that for $r\in \mathbb{N}$, if $n$ is sufficiently large, then any $r$-colouring of $[n]$ results in monochromatic $a,b,c\in [n]$ such that $ab=c$. In this paper we are interested in the…

We generalize a result of Tibor Gallai as follows: for any finite set of points $\mathcal{S}$ in the plane, if the plane is colored in finitely many colors, then there exist $2^{\aleph_0}$ monochromatic subsets of the plane homothetic to…

A well-known result by Graham in Euclidean Ramsey Theory states that, for every positive real number $A$, every coloring of the plane with finite number of colors contains a monochromatic triangle of area $A$. We consider canonical versions…

We show that for $m, r \in \mathbb{N}$ and $N > (2m+1)^r (r!)^{1/m}$, every $r$-coloring of the integers in the interval $[N]$ contains a monochromatic solution to the equation \[ x_1 + \dots + \dots x_{m+1} = y_1 + \dots + y_m. \] This…

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k-1)-regular…

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…

We prove a canonical polynomial Van der Waerden's Theorem. More precisely, we show the following. Let $\{p_1(x),\ldots,p_k(x)\}$ be a set of polynomials such that $p_i(x)\in \mathbb{Z}[x]$ and $p_i(0)=0$, for every $i\in \{1,\ldots,k\}$.…

Let $P$ be a set of $n$ points in the plane, not all on a line, each colored \emph{red} or \emph{blue}. The classical Motzkin--Rabin theorem guarantees the existence of a \emph{monochromatic} line. Motivated by the seminal work of Green and…

Let $n\in \mathbb{N}$, $R$ be a binary relation on $[n]$, and $C_1(i,j),\ldots,C_n(i,j) \in \mathbb{Z}$, for $i,j \in [n]$. We define the exponential system of equations $\mathcal{E}(R,(C_k(i,j)_{i,j,k})$ to be the system \[…

The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C in R^d of cardinality (d+1)t, partitioned into t-point subsets C_1,C_2,...,C_{d+1} (which we think of as color classes; e.g., the…