INTEGERS 14 (2014) #A31 PDF De and Hindman proposed that a particular system should be partition regular but not partition regular near zero. With Neil Hindman and Imre Leader I found a different example; in this paper I show that De and Hindman’s original system also works.
Category: Bibliography
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 …
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 …
A note on balanced independent sets in the cube
Australas. J. Combin. 52 (2012), 205–207. PDF How large can an independent set in the discrete cube be if it contains equal numbers of sets of even and odd size? Take odd sets starting from the bottom of the cube, and even sets starting from the top. Proving that this works uses an isoperimetric inequality: …