Partition regularity without the columns property

Ben Barber, Neil Hindman, Imre Leader and Dona Strauss, Proc. Amer. Math. Soc. 143 (2015), 3387-3399 PDF

Rado's theorem states that a finite matrix is partition regular if and only if it has the "columns property". It is easy to write down infinite matrices with the columns property that are not partition regular, but all known examples of partition regular matrices do have the columns property. In this paper we describe a matrix that is partition regular but fails to have the columns property in the strongest possible sense.

The main contribution of this paper is a translation of the key lemma of "Partition regularity in the rationals" to work with central, rather than dense, sets. Central sets have very strong combinatorial properties; for example, they contain solutions to all finite partition regular systems. As a result, our theorems are harder to prove but easier to apply—for the application above we could have proved the partition regularity of the systems using density methods, but the argument would have been more involved.

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