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Image Partition Regularity of Matrices

  • Neil Hindman (a1) and Imre Leader (a2)

Abstract

Many of the classical results of Ramsey Theory, including those of Hilbert, Schur, and van der Waerden, are naturally stated as instances of the following problem: given a u × ν matrix A with rational entries, is it true, that whenever the set ℕ of positive integers is finitely coloured, there must exist some x∈ℕν such that all entries of Ax are the same colour? While the theorems cited are all consequences of Rado's theorem, the general problem had remained open. We provide here several solutions for the alternate problem, which asks that x∈ℕν. Based on this, we solve the general problem, giving various equivalent characterizations.

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[1]Bergelson, V., Deuber, W. and Hindman, N. (1990) Nonmetrizable topological dynamics and Ramsey Theory. Trans. Amer. Math. Soc. 320 293320.
[2]Deuber, W. (1973) Partitionen und lineare Gleichungssysteme. Math. Zeit. 133 109123.
[3]Furstenberg, H. (1981) Recurrence in ergodic theory and combinatorial number theory, Princeton Univ. Press, Princeton.
[4]Hilbert, D. (1982) Über die Irreducibilität ganzer Rationaler Funktionen mit ganzzahligen Koeffizienten. J. Reine Angew Math. 110 104129.
[5]Hindman, N. and Woan, W. (to appear) Central sets in semigroups and partition regularity of systems of linear equations. Mathematika.
[6]Rado, R. (1933) Studien zur Kombinatorik. Math. Zeit. 36 424480.
[7]Schur, I. (1916) Über die Kongruenz xm + ym = zm(mod p). Jahresbericht der Deutschen Math. – Verein. 25 114117.
[8]van der Waerden, B. (1927) Beweis einer Baudetschen Vermutung. Nieuw Arch. Wisk. 15 212216.

Image Partition Regularity of Matrices

  • Neil Hindman (a1) and Imre Leader (a2)

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