Partition regularity and other combinatorial problems

This is the imaginative title of my PhD thesis.  It contains four unrelated pieces of work.  (I was warned off using this phrasing in the thesis itself, where the chapters are instead described as “self-contained”.) The first and most substantial concerns partition regularity.  It is a coherent presentation of all of the material from Partition …

Nowhere zero 6-flows

A flow on a graph is an assignment to each edge of of a direction and a non-negative integer (the flow in that edge) such that the flows into and out of each vertex agree.  A flow is nowhere zero if every edge is carrying a positive flow and (confusingly) it is a -flow if the flows …

Fractional clique decompositions of dense graphs and hypergraphs

Ben Barber, Daniela Kühn, Allan Lo, Richard Montgomery and Deryk Osthus, Journal of Combinatorial Theory, Series B, Volume 127, November 2017, Pages 148–186 PDF Together with Daniela Kühn, Allan Lo and Deryk Osthus I proved that for every graph there is a constant such that every “-divisible” graph on vertices with minimum degree at least has an …

Edge-decompositions of graphs with high minimum degree

Ben Barber, Daniela Kühn, Allan Lo and Deryk Osthus, Advances in Mathematics, Volume 288, 22 January 2016, Pages 337–385 PDF When can the edge set of a graph be partitioned into triangles? Two obvious necessary conditions are that the total number of edges is divisible by 3 and the degree of every vertex is even. We …

Distinguishing subgroups of the rationals by their Ramsey properties

Ben Barber, , Neil Hindman, Imre Leader and Dona Strauss, Journal of Combinatorial Theory, Series A, Volume 129, January 2015, Pages 93–104 PDF In Partition regularity in the rationals we (Barber, Hindman and Leader) showed that there are systems of equations that are partition regular over but not over . Here we show that this separation is …

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 …

Partition regularity of a system of De and Hindman

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.

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 …