Partition regularity with congruence conditions

Ben Barber and Imre Leader, Journal of Combinatorics, Volume 4 (2013), Number 3 PDF

Does a partition regular system remain partition regular if we ask that each variable is divisible by ? Not necessarily. This answers several open questions from Hindman, Leader and Strauss's 2003 survey.

The proof of Proposition 5 in the journal version is not entirely clear; I recommend reading the updated pdf linked above. My thanks to Boaz Tsaban for pointing this out.

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