Multiply partition regular matrices

作者：Davenport Dennis; Hindman Neil^*; Leader Imre; Strauss Dona

来源：Discrete Mathematics, 2014, 322: 61-68.

DOI：10.1016/j.disc.2014.01.003

• Matrix
• Image partition regular
• Kernel partition regular
• Columns condition
• Rado%26apos
• s theorem

Let A be a finite matrix with rational entries. We say that A is doubly image partition regular if whenever the set N of positive integers is finitely coloured, there exists (x) over right arrow such that the entries of A (x) over right arrow are all the same colour (or monochromatic) and also, the entries of (x) over right arrow are monochromatic. Which matrices are doubly image partition regular? %26lt;br%26gt;More generally, we say that a pair of matrices (A, B), where A and B have the same number of rows, is doubly kernel partition regular if whenever N is finitely coloured, there exist vectors (x) over right arrow and (y) over right arrow, each monochromatic, such that A (x) over right arrow + B (y) over right arrow = (0) over right arrow. (So the case above is the case when B is the negative of the identity matrix.) There is an obvious sufficient condition for the pair (A, B) to be doubly kernel partition regular, namely that there exists a positive rational c such that the matrix M = (A cB) is kernel partition regular. (That is, whenever N is finitely coloured, there exists monochromatic (x) over right arrow such that M (x) over right arrow = (0) over right arrow.) Our aim in this paper is to show that this sufficient condition is also necessary. As a consequence we have that a matrix A is doubly image partition regular if and only if there is a positive rational c such that the matrix (A -cI) is kernel partition regular, where I is the identity matrix of the appropriate size. %26lt;br%26gt;We also prove extensions to the case of several matrices.

• 出版日期2014-5-6