Related papers: Partition regularity without the columns property

We study the row completion problem of polynomial and rational matrices with partial prescription of the structural data. The prescription of the complete structural data has been solved in Amparan et al., Lin. Alg. Appl. 720 (2025)…

This paper considers the problem of matrix completion when some number of the columns are completely and arbitrarily corrupted, potentially by a malicious adversary. It is well-known that standard algorithms for matrix completion can return…

Since the theorems of Schur and van der Waerden, numerous partition regularity results have been proved for linear equations, but progress has been scarce for non-linear ones, the hardest case being equations in three variables. We prove…

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to $1$. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if $d$…

A colored complete graph is said to be Gallai-colored if it contains no rainbow triangle. This property has been shown to be equivalent to the existence of a partition of the vertices (of every induced subgraph) in which at most two colors…

Raimi's classical theorem establishes a partition of the natural numbers with a remarkable unavoidability property: for every finite coloring of $\mathbb{N}$, there is a color class whose translate meets both parts of the partition in…

In this paper, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations of the entries in the first row with integer…

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…

In this article, we introduce the notion of regular fusible modules. Let $R$ be a ring with an identity and $M$ an $R$-module. An element $0\neq m\in M$ is said to be regular fusible if there exists $r\in R$, a non zero-divisor of $M$, such…

Call a noncommutative rational function $r$ regular if it has no singularities, i.e., $r(X)$ is defined for all tuples of self-adjoint matrices $X$. In this article regular noncommutative rational functions $r$ are characterized via the…

We study a class of square matrices with non-negative elements which have cyclically monotone rows in the sense that each row of a matrix from the class consists of a cyclically non-increasing sequence of numbers starting from a maximal…

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…

We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it similar to a direct sum of all $1's$ matrices and a 0 matrix via a unitary monomial similarity. In particular, the only…

In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over…

Recently, Hirschhorn and Sellers defined the partition function $a_r(n)$, which counts the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may appear in one of $r$-colors for fixed $r\ge1$. The aim…

In the Stable Roommates problem, we seek a stable matching of the agents into pairs, in which no two agents have an incentive to deviate from their assignment. It is well known that a stable matching is unlikely to exist, but a stable…

Finite dimensional matrices having more columns than rows have no left inverses while those having more rows than columns have no right inverses. We give generalizations of these simple facts to bi--infinite matrices and use those to obtain…

An integer-valued matrix $\mathbf{A}$ is $\Delta$-modular if each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix has determinant at most $\Delta$ in absolute value. The column number problem is to determine the maximum…

A vertex colouring of some graph is called perfect if each vertex of colour $i$ has exactly $a_{ij}$ neighbours of colour $j$. Being perfect imposes several restrictions on the colour incidence matrix $(a_{ij})$. We list several (old and…

Given a finite coloring (or finite partition) of the free semigroup $A^+$ over a set $A$, we consider various types of monochromatic factorizations of right sided infinite words $x\in A^\omega$. Some stronger versions of the usual notion of…