Partition regularity in the rationals

Abstract

A system of homogeneous linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, the system has a monochromatic solution. The Finite Sums theorem provided the first example of an infinite partition regular system of equations. Since then, other such systems of equations have been found, but each can be viewed as a modification of the Finite Sums theorem. We present here a new infinite partition regular system of equations that appears to arise in a genuinely different way. This is the first example of a partition regular system in which a variable occurs with unbounded coefficients. A modification of the system provides an example of a system that is partition regular over Q but not N, settling another open problem.Comment: 11 page

Partition regularity without the columns property

“…Now suppose we have a system x n + 2x n+1 = y n in R for n < ω such that {x n : n < ω} ∪ {y n : n < ω} is monochromatic with respect to γ. If for any n we have b(x n+1 ) > b(x n ), then we get b(y n ) = b(x n+1 ) and α y n , b(y n ) = 2α x n+1 , b(x n+1 ) , contradicting requirement (2). Therefore, for each n, b(x n+1 ) ≤ b(x n ).…”

“…The proof that the system mentioned above is partition regular over N depends on the following lemma, which is based on results in [2]. The lemma refers to central sets.…”

Partition regularity with congruence conditions

“…[The result in [1] was stated for c n = 2 n , but, as remarked in [1], the proof works for any integer sequence (c n ). ]…”

“…For example, Schur's theorem states that whenever we finitely colour the natural numbers we can find x and y such that x, y and x + y are all the same colour: that is, the matrix   1 0 0 1 1 1   is image partition regular. 1 In the finite case, partition regularity is very well understood (see [5], [4] and, for a general overview, [2]). In contrast, in the infinite case the situation is far less clear, and several very basic questions remain unanswered-see the survey [3].…”

“…Finally, we say that an integer matrix A (with only finitely many nonzero entries in each row) is image partition regular over Q if, whenever Q \ {0} is finitely coloured, there is a vector x with entries in Q such that Ax is monochromatic. The corresponding question for kernel partition regularity was answered in [1].…”