Consistency for partition regular equations
نویسندگان
چکیده
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,...
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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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An n ×m rational matrix A is said to be partition regular if for every finite coloring of N there is a monochromatic vector ~x ∈ N with A~x = ~0. A set D ⊆ N is said to be partition regular for A (or for the system of equations A~x = ~0) if for every finite coloring of D there is a monochromatic ~x ∈ D with A~x = ~0. In this paper we show that for every n there is a set that is partition regula...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2006
ISSN: 0012-365X
DOI: 10.1016/j.disc.2005.10.030