2013/01/08 by Baglini, Lorenzo Luperi · 1 citation
#03H05 #05D10 #12L15 #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1301.1467
In 2011, Neil Hindman proved that for every natural number n,m the polynomial ∑i=1n xi-∏j=1m 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)=∑i=1naixiMi(y1,...,ym), where n,m are natural numbers, ∑i=1naixi 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 chosing the appropriate nonstandard setting, the proof of the main results can be obtained by very simple algebraic considerations.