Ramsey properties of nonlinear Diophantine equations

Abstract

We prove general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of nonlinear equations. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters βN, grounding on combinatorial properties of positive density sets and IP sets. Necessary conditions are proved by a new technique in nonstandard analysis, based on the use of the relation of u-equivalence for the hypernatural numbers * N.

Partition regularity of polynomial systems near zero

“…In this paper, we showed that Theorem 4 could be seen as a particular case of a more general characterization of partition regular equations 5 , that we proved by means of nonstandard methods. The same result is proven in [3] by means of purely standard methods based on ultrafilters. To introduce the result, we first need a definition: Definition 3 Let m be a positive natural number and let {y 1 , .…”

“…Proof This result can be proven following the same ideas of the proof of Theorem 3.3 in [9], if one wants to use nonstandard methods, or of Theorem 2.10 in [3], if one wants to use purely standard arguments based on ultrafilters. We adapt here the proof of Theorem 2.10 in [3] (which talked about the partition regularity on N and was more complicated as we handled also the injectivity properties of the sets of solutions of equation 3) to our present case. Let U be an idempotent ultrafilter in K (β(0, 1), ) or K (β S, ).…”

“…The polynomial a + b − cd is not homogeneous, so its partition regularity near zero can not be directly deduced from Theorem 10 and its corollaries. However, we can prove that it is partition regular near zero (and construct ultrafilters that witness its partition regularity) following the methods first introduced in [9], using nonstandard analysis, and then developed also in [3] by purely standard methods. We will use the following known simple fact (which is a trivial consequence of idempotency, see for example [7,Theorem 5.12]): Proposition 3 Let (S, •) be a subsemigroup of R + , • and let U be an idempotent in (β S, ).…”

“…Let us notice that an immediate, but interesting, consequence of Theorem 2 is the following 3 : Theorem 3 Let u, v ∈ N. Let A be a u × v matrix with entries from R, with columns c 1 , . .…”

“…Several other results about the partition regularity of equations on N have been proven in the past few years; we refer to [3] for an overview, and to [12] for the latest results we are aware of.…”