منابع مشابه
Consistency for partition regular equations
It is easy to deduce from Ramsey’s Theorem that, given positive integers a1, a2, . . . , am and a finite colouring of the set N of positive integers, there exists an injective sequence (xi) ∞ i=1 with all sums of the form ∑m i=1 aixri (r1 < r2 < · · · < rm) lying in the same colour class. The consistency version of this result, namely that, given positive integers a1, a2, . . . , am and b1, b2,...
متن کاملIndependence for partition regular equations
A matrix A is said to be partition regular (PR) over a subset S of the positive integers if whenever S is finitely coloured, there exists a vector x, with all elements in the same colour class in S, which satisfies Ax = 0. We also say that S is PR for A. Many of the classical theorems of Ramsey Theory, such as van der Waerden’s Theorem and Schur’s Theorem, may naturally be interpreted as statem...
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3 (m,p,c)-sets 59 4 Canonical Results 63 5 Coloring Objects of Higher Rank 65 1 Introduction In 1930 Ramsey published his paper On a problem in formal logic 12]. He established a result, nowadays known as Ramsey's Theorem: Let k and r be positive integers. Then for every r-coloring of the k-element subsets of ! there exists an innnite subset S ! such that all k-element subsets of S are colored ...
متن کاملExtensions of Infinite Partition Regular Systems
A finite or infinite matrixA with rational entries (and only finitely many non-zero entries in each row) is called image partition regular if, whenever the natural numbers are finitely coloured, there is a vector x, with entries in the natural numbers, such that Ax is monochromatic. Many of the classical results of Ramsey theory are naturally stated in terms of image partition regularity. Our a...
متن کاملCompactness of Systems of Equations on Completely Regular Semigroups
A semigroup S is said to have the compactness property, or CP for short, if each system of equations over a nite set of variables has an equivalent nite subsystem, that is, having exactly the same solutions in S. We prove that a completely 0-simple semigroup S satis es CP if and only if the group G in a Rees matrix representation S = M0(I; G; ;P) satis es this property. Further, a variety of co...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 1991
ISSN: 0097-3165
DOI: 10.1016/0097-3165(91)90072-o