Paul A. Russell

Independence for partition regular equations

A matrix A is said to be partition regular (PR) over a subset S of the positive integers if whenever S is finitely coloured, there exists a vector x, with all elements in the same colour class in S, which satisfies Ax=0. We also say that S is PR for A. Many of the classical theorems of Ramsey Theory, such as van der Waerdens theorem and Schurs theorem, may naturally be interpreted as statements about partition regularity. Those matrices which are partition regular over the positive integers were completely characterised by Rado in 1933. Given matrices A and B, we say that A Rado-dominates B if any set which is PR for A is also PR for B. One trivial way for this to happen is if every solution to Ax=0 actually contains a solution to By=0. Bergelson, Hindman and Leader conjectured that this is the only way in which one matrix can Rado-dominate another. In this paper, we prove this conjecture for the first interesting case, namely for 1x3 matrices. We also show that, surprisingly, the conjecture is not true in general.

Sparse partition regularity

Our aim in this paper is to prove Deubers conjecture on sparse partition regularity, that for every

Compressions and probably intersecting families

m

Transitive sets in Euclidean Ramsey theory

,

Probably intersecting families are not nested

p

Transitive Sets and Cyclic Quadrilaterals

and

Hamilton Paths in Certain Arithmetic Graphs.

c

Note: Families intersecting on an interval

there exists a subset of the natural numbers whose

Inhomogeneous Partition Regularity.

(m,p,c)

Infinite monochromatic sumsets for colourings of the reals

-sets have high girth and chromatic number. More precisely, we show that for any