Related papers: Central sets and infinite monochromatic exponentia…
We prove that for any coloring of the naturals using two colors there are monochromatic sets of the form $\{x,y,xy,x+iy:i\leq k\}$ and $\{x,y,x^y,xy^i:i\leq k\}$ for any $k$.
Recall that van der Waerden's theorem states that any finite coloring of the naturals has arbitrarily long monochromatic arithmetic sequences. We explore questions about the set of differences of those sequences.
We prove upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem. For example, we show that Hindman's Theorem for sums of length at most 2 and 4 colors…
W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left\{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\right\} $ by showing that for any finite coloring of $X_{k}$ there exists a sequence such that the Gower subspace…
An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of non-linear…
We show that there is a rational vector space $V$ such that, whenever $V$ is finitely coloured, there is an infinite set $X$ whose sumset $X+X$ is monochromatic. Our example is the rational vector space of dimension…
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 \[…
Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…
Suppose that we have a finite colouring of the reals. What sumset-type structures can we hope to find in some colour class? One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums…
We take a unified approach to central limit theorems for a class of irreducible urn models with constant replacement matrix. Depending on the eigenvalue, we consider appropriate linear combinations of the number of balls of different…
Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open…
Inspired by a question of Kra, Moreira, Richter, and Robertson, we prove two new results about infinite polynomial configurations in large subsets of the rational numbers. First, given a finite coloring of $\mathbb{Q}$, we show that there…
We show that for every coloring of the rationals into finitely many colors, one of the colors contains a set of the form $\{x,y,xy,x+y\}$ for some nonzero $x$ and $y$.
We show that a finite coloring of an amenable group contains `many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables. This gives the first combinatorial proof and extensions of Bergelson and…
We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite…
For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$…
Given a coloring of the k-element subsets of an uncountable separable metric space, we show that there exists an infinite monochromatic subset which contains its limit point.
We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let $\mathsf{HT}^{\leq n}_k$ denote the assertion…
Hindman's Theorem (HT) states that for every coloring of $\mathbb N$ with finitely many colors, there is an infinite set $H \subseteq \mathbb N$ such that all nonempty sums of distinct elements of $H$ have the same color. The investigation…
We prove a theorem ensuring that the compositions of certain Ramsey families are still Ramsey. As an application, we show that in any finite coloring of $\mathbb{N}$ there is an infinite set $A$ and an as large as desired finite set $B$…