Rado's criterion over squares and higher powers

Abstract

We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equation remains valid over the set of positive kth powers, provided the equation has at least (1 + o(1))k log k variables. We thus completely describe which diagonal forms are partition regular and which are not, given sufficiently many variables. In addition, we prove a supersaturated version of Rado's theorem for a linear equation restricted either to squares minus one or to logarithmically-smooth numbers.Part 2. Rado's criterion over squares and higher powers 9. The smooth homogeneous Bergelson-Leibman theorem 10. A supersaturated generalisation of both Roth and Sárközy's theorems 11. Pseudorandom Roth-Sárközy 12. The W -trick for smooth powers and a non-linear Roth-Sárközy theorem 13. Deducing partition regularity

Polynomial Schur's theorem

“…Ramsey theory has witnessed exciting development recently. We refer the readers to the papers of Green and Sanders [7] and of Moreira [12] for the problem involving sum and product of x and y, and to the paper of Chow, Lindqvist and Prendiville [3] for generalisation of Rados criterion to higher powers for sufficiently many variables.…”

“…Let φ be a 2-colouring of [n]. Then the number of monochromatic solutions {x, y, z} ∈ [n] (3) to x + y = p(z) is at least n 2/d 3 −o (1) . Moreover, there is a 2-colouring for which the number of monochromatic solutions is only O(n 2/d 2 ).…”

Partition regularity for systems of diagonal equations

“…Recent developments in the study of partition regularity are motivated by the desire to extend Rado's classification to systems of non-linear equations. Chow, Lindqvist, and Prendiville [CLP18] recently established the following generalisation of Rado's criterion for diagonal polynomial equations in sufficiently many variables.…”

“…For this reason, we are required to impose some form of non-singularity condition on our system, such as (I), so that we may use the Hardy-Littlewood circle method to count solutions. This allows us to develop the tools introduced by Chow, Lindqvist, and Prendiville [CLP18] so that we may establish partition regularity for systems of equations.…”

“…In §4 we introduce the notion of multiplicative syndeticity, and recall the induction on colours argument from [CLP18]. This allows us to reduce both Theorem 1.2 and Theorem 1.4 to a result, Theorem 4.5, concerning solutions of (1.2) over dense and multiplicatively syndetic sets.…”