Fermat-like equations that are not partition regular

Ramsey properties of nonlinear Diophantine equations

“…Example 3.18. In [12], it is proved that the following polynomials x n +y m = z k are not PR for k / ∈ {n, m}. This result is obtained as a particular case of Corollary 3.12.…”

“…To our knowledge, the last progress done in this area about is found in [12], where M. Riggio and the first named author used nonstandard analysis to identify a large class of Fermat-like equations that are not partition regular, the simplest examples being x m + y n = z k where k / ∈ {n, m}. 3 At the moment this paper was completed, it was breaking news that M. J. H. Heule, O. Kullmann and V. W. Marek [18] solved a problem posed by P. Erdős and R. Graham in the 1970s, namely the Boolean Pythagorean triples problem, that asked whether the equation x 2 + y 2 = z 2 is partition regular for 2-colorings of N. By using a computer-assisted proof, they have been able to prove that any 2-coloring of {1, 2, .…”

“…The polynomial P (x 1 , x 2 , x 3 ) = x 1 x 2 −2x 3 is PR but it does not belong to the family F of Theorem 2.8. 12 Indeed, given a finite coloring…”

“…In the last years, the interest on problems related to the partition regularity of nonlinear Diophantine equations has been rising constantly (see, e.g., [28,23,6,37,33,34,3,12,13,18]). We hope that this paper will contribute to a general Ramsey theory of nonlinear Diophantine equations.…”