Partition regulairty of nonlinear polynomials: a nonstandard approach

In 2011, Neil Hindman proved that for all natural numbers n, m the polynomial Pn i=1 xi Q m j=1 yj has monochromatic solutions for every finite coloration of N. We want to generalize this result to two classes of nonlinear polynomials. The first class consists of polynomials P(x1, . . . , xn, y1, . . . , ym) of the following kind: P(x1, . . . , xn, y1, . . . , ym) = Pn i=1 aixiMi(y1, . . . , ym), where n, m are natural numbers, Pn i=1 aixi has monochromatic solutions for every finite coloration of N and the degree of each variable y1, . . . , ym in Mi(y1, . . . , ym) is at most one. An example of such a polynomial is x1y1 + x2y1y2 x3. The second class of polynomials generalizing Hindman’s result is more complicated to describe; its particularity is that the degree of some of the involved variables can be greater than one. The technique that we use relies on an approach to ultrafilters based on Nonstandard Analysis. Perhaps, the most interesting aspect of this technique is that, by carefully choosing the appropriate nonstandard setting, the proof of the main results can be obtained by very simple algebraic considerations.