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JCT

2007

2007

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 statements about partition regularity. Those matrices which are partition regular over the positive integers were completely characterized by Rado in 1933. Given matrices A and B, we say that A Rado-dominates B if any set which is PR for A is also PR for B. One trivial way for this to happen is if every solution to Ax = 0 actually contains a solution to By = 0. Bergelson, Hindman and Leader conjectured that this is the only way in which one matrix can Rado-dominate another. In this paper, we prove this conjecture for the ﬁrst interesting case, namely for 1 × 3 matrices. We also show that, sur...

Related Content

Added |
15 Dec 2010 |

Updated |
15 Dec 2010 |

Type |
Journal |

Year |
2007 |

Where |
JCT |

Authors |
Imre Leader, Paul A. Russell |

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