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Register a new userPartition regularity without the columns property
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A finite or infinite matrix A with rational entries is called partition regular if whenever the natural numbers are finitely coloured there is a monochromatic vector x with Ax=0. Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular. In the finite case, Rado proved that a matrix is partition regular if and only it satisfies a computable condition known as the columns property. The first requirement of the columns property is that some set of columns sums to zero. In the infinite case, much less is known. There are many examples of matrices with the columns property that are not partition regular, but until now all known examples of partition regular matrices did have the columns property. Our main aim in this paper is to show that, perhaps surprisingly, there are infinite partition regular matrices without the columns property --- in fact, having no set of columns summing to zero. We also make a conjecture that if a partition regular matrix (say with integer coefficients) has bounded row sums then it must have the columns property, and prove a first step towards this.
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We say that the system of equations $Ax=b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax=b$. Rado proved that the system $Ax=b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new `direct proof of Rados result.
In cite{dehind1}, the concept of image partition regularity near zero was first instigated. In contrast to the finite case , infinite image partition regular matrices near zero are very fascinating to analyze. In this regard the abstraction of Centrally image partition regular matrices near zero was introduced in cite{biswaspaul}. In this paper we propose the notion of matrices that are C-image partition regular near zero for dense subsemigropus of $((0,infty),+)$.
A Cayley graph is said to be an NNN-graph if it is both normal and non-normal for isomorphic regular groups, and a group has the NNN-property if there exists an NNN-graph for it. In this paper we investigate the NNN-property of cyclic groups, and show that cyclic groups do not have the NNN-property.
We introduce the set $mathcal{G}^{rm SSP}$ of all simple graphs $G$ with the property that each symmetric matrix corresponding to a graph $G in mathcal{G}^{rm SSP}$ has the strong spectral property. We find several families of graphs in $mathcal{G}^{rm SSP}$ and, in particular, characterise the trees in $mathcal{G}^{rm SSP}$.
Let $H$ be connected $m$-uniform hypergraph and $mathcal{A}(H)$ be the adjacency tensor of $H$. The stabilizing index of $H$, denoted by $s(H)$, is exactly the number of eigenvectors of $mathcal{A}(H)$ associated with the spectral radius, and the cyclic index of $H$, denoted by $c(H)$, is the number of eigenvalues of $mathcal{A}(H)$ with modulus equal to the spectral radius. Let $bar{H}$ be a $k$-fold covering of $H$. Then $bar{H}$ is isomorphic to a hypergraph $H_B^phi$ derived from the incidence graph $B_H$ of $H$ together with a permutation voltage assignment $phi$ in the symmetric group $mathbb{S}_k$. In this paper, we first characterize the connectedness of $bar{H}$ by using $H_B^phi$ for subsequent discussion. By applying the theory of module and group representation, we prove that if $bar{H}$ is connected, then $s(H) mid s(bar{H})$ and $c(H) mid c(bar{H})$. In the situation that $bar{H}$ is a $2$-fold covering of $H$, if $m$ is even, we show that regardless of multiplicities, the spectrum of $mathcal{A}(bar{H})$ contains the spectrum of $mathcal{A}(H)$ and the spectrum of a signed hypergraph constructed from $H$ and the covering projection; if $m$ is odd, we give an explicit formula for $s(bar{H})$.
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