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Rado's Theorem characterizes the systems of homogenous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions…
We prove general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of nonlinear equations. Sufficient conditions are obtained by…
By using nonstandard analysis, and in particular iterated hyper-extensions, we give foundations to a peculiar way of manipulating ultrafilters on the natural numbers and their pseudo-sums. The resulting formalism is suitable for…
In recent years, several problems regarding the partition regularity of exponential configurations have been studied in the literature, in some cases using the properties of specific ultrafilters. In this paper, we start to lay down the…
We study Rado functionals and the maximal condition (first introduced by J. M. Barret et al.) in terms of the partition regularity of mixed systems of linear equations and inequalities. By strengthening the maximal Rado condition, we…
We say that a list $\Lambda =\{ \lambda _{1},\ldots ,\lambda _{n}\}$ of complex numbers is realizable, if it is the spectrum of a nonnegative matrix $A$ (the realizing matrix). We say that $\Lambda $ is universally realizable if it is…
We consider the equation $$ ab + cd = \lambda, \qquad a\in A, b \in B, c\in C, d \in D, $$ over a finite field $F_q$ of $q$ elements, with variables from arbitrary sets $ A, B, C, D \subseteq F_q$. The question of solvability of such and…
Ultrafilters are very useful and versatile objects with applications throughout mathematics: in topology, analysis, combinarotics, model theory, and even theory of social choice. Proofs based on ultrafilters tend to be shorter and more…
A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to…
We consider Rado numbers of the regular equations $\mathcal{E}(b)$ of the form \[ c_1x_1+c_2x_2+\dots+ c_{k-1}x_{k-1} = x_k + b, \] where $b \in \mathbb{Z}$ and $c_i \in \mathbb{Z}^{+}$ for all $i$. We give the upper bounds and the…
Let $a_1,\ldots,a_m$ be nonzero integers, $c \in \mathbb Z$ and $r \ge 2$. The Rado number for the equation \[ \sum_{i=1}^m a_ix_i = c \] in $r$ colours is the least positive integer $N$ such that any $r$-colouring of the integers in the…
We extend Deuber's theorem on $(m,p,c)$-sets to hold over the multidimensional positive integer lattices. This leads to a multidimensional Rado theorem where we are guaranteed monochromatic multidimensional points in all finite colorings of…
We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show…
We give complete and exact descriptions of spaces of ultradifferentiable functions that are closed under composition with either holomorphic or ultradifferentiable functions -- which are two distinct cases. The proof works by considering…
We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters. We also list some problems, and furnish applications to topological spaces and to extended logics.
For ultrafilters u,v on N, the operation u/v is introduced and formalised which acts as quotient-like structures when v strongly divides u.Central to our study is the characterization of self-divisible ultrafilters in connection with the…
We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equation remains valid over the set of positive $k$th powers,…
In this paper, we set up a general correspondence between the algebra properties of $\bN$ and the sets defined by dynamical properties. In particular, we obtain a dynamical characterization of C-sets, where C-sets are the sets satisfying…
Linear vector equations and inequalities are considered defined in terms of idempotent mathematics. To solve the equations, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces. The…
We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers…