A canonical version for partition regular systems of linear equations
نویسندگان
چکیده
منابع مشابه
Partition Regular Systems of Linear Inequalities
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 ...
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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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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 1986
ISSN: 0097-3165
DOI: 10.1016/0097-3165(86)90117-2