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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 $((0,infty),+)$.
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We give a general method to reduce Hurewicz-type selection hypotheses into standard ones. The method covers the known results of this kind and gives some new ones. Building on that, we show how to derive Ramsey theoretic characterizations for these selection hypotheses.
Image partition regular matrices near zero generalizes many classical results of Ram- sey Theory. There are several characterizations of finite image partition regular matrices near zero. Contrast to the finite cases there are only few classes of matri- ces that are known to be infinite image partition regular near zero. In this present work we have produced several new examples of such classes.
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.
A finite or infinite matrix $A$ is image partition regular provided that whenever $mathbb{N}$ is finitely colored, there must be some $overset{rightarrow}{x}$ with entries from $mathbb{N}$ such that all entries of $A overset{rightarrow}{x}$ are in the same color class. Comparing to the finite case, infinite image partition regular matrices seem more harder to analyze. The concept of centrally image partition regular matrices were introduced to extend the results of finite image partition regular matrices to infinite one. In this paper, we shall introduce the notion of C-image partition regular matrices, an interesting subclass of centrally image partition regular matrices. Also we shall see that many of known centrally image partition regular matrices are C-image partition regular.
Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of $((0,infty),+)$ and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-$breve{C}$ech compactification, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central theorem near zero. Algebraically one can define quasi-central set near zero for dense subsemigroup of $((0,infty),+)$, and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of $((0,infty),+)$, C-sets near zero are the sets, which satisfies the conclusions of the central sets theorem near zero. Like discrete case, we shall produce dynamical characterizations of these combinatorically rich sets near zero.
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