Related papers: Polynomial Schur's theorem

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph…

Let $Q_n$ be the poset that consists of all subsets of a fixed $n$-element set, ordered by set inclusion. The poset cube Ramsey number $R(Q_n,Q_n)$ is defined as the least $m$ such that any 2-coloring of the elements of $Q_m$ admits a…

A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the…

In 2008, M. Marshall settled a long-standing open problem by showing that if f(x,y) is a polynomial that is non-negative on the strip [0,1] x R, then there exist sums of squares s(x,y) and t(x,y) such that f(x,y) = s(x,y) + (x - x^2)…

Consider the $n$th degree polynomial equation, $X^n+A_{n-1}X^{n-1}+...+A_1X+A_0=0$ over the ring of 2 by 2 complex matrices. If this equation has more than ${2n \choose 2}$ solutions, then it has infinitely many solutions. We show here that…

Ramsey's theorem asserts that every $k$-coloring of $[\omega]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$, there exists a computable $k$-coloring of $[\omega]^n$ whose solutions compute the halting set. On the other hand,…

We consider Dirichlet elliptic equations driven by the sum of a $p$-Laplacian $(2<p)$ and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both $\pm\infty$ and at zero. We prove an existence theorem…

We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.

Let $P_t$ denote the path on $t$ vertices. The $r$-coloured Ramsey number of $P_t$, denoted by $R_r(P_t)$, is the minimum integer $n$ such that whenever the complete graph on $n$ vertices is given an $r$-edge-colouring, there exists a…

We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to approximation by…

The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby,…

We show, assuming Schanuel's conjecture, that every irreducible complex polynomial in two variables where both variables appear has infinitely many algebraically independent solutions of the form (z,e^z).

We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$…

In this paper we consider polynomial representability of functions defined over $Z_{p^n}$, where $p$ is a prime and $n$ is a positive integer. Our aim is to provide an algorithmic characterization that (i) answers the decision problem: to…

For integers $k,r\geq 2$, the diagonal Ramsey number $R_r(k)$ is the minimum $N\in\mathbb{N}$ such that every $r$-coloring of the edges of a complete graph on $N$ vertices yields on a monochromatic subgraph on $k$ vertices. Here we make a…

We prove new cases of reasonable bounds for the polynomial Szemer\'{e}di theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemer\'edi's theorem in the integers…

A multivariate polynomial is {\em stable} if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra $\A_n$ that preserve stability. An important…

The Hales--Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales--Jewett numbers are primitive recursive. A key tool used in his proof, now known as the…

The two-colour Ramsey number $R(m,n)$ is the least natural number $p$ such that any graph of order $p$ must contain either a clique of size $m$ or an independent set of size $n$. We exhibit a method for computing upper bounds for $R(m,n)$…

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…