Partition regularity in the rationals

Ben Barber, Neil Hindman, Imre Leader, Journal of Combinatorial Theory, Series A, Volume 120, Issue 7, September 2013, Pages 1590–1599 PDF A system of linear equations is partition regular if, whenever the natural numbers are finitely coloured, the system of equations has a monochromatic solution. Partition regularity can also be defined over the rationals, and …

Random walks on quasirandom graphs

Ben Barber and Eoin Long, The Electronic Journal of Combinatorics, 20(4) (2013), #P25 PDF Take a long (proportional to ) random walk in a quasirandom graph . Must the subgraph of edges traversed by be quasirandom? We’d like to say yes, for the following reason: visits every vertex about the same number of times, so we …

Piecewise syndetic and van der Waerden

Joel Moreira has just proved that whenever the natural numbers are finitely coloured we can find and such that , and are all the same colour.  He actually proves a much more general result via links to topological dynamics, but he includes a direct proof of this special case assuming only a consequence of van der Waerden’s …