This was the focus of a Summer Research Scholarship that I held in the summer of 2018/19, under the supervision of Martino Lupini. We attempted to find general necessary and sufficient conditions to prove / disprove the following property of a Diophantine equation:
Definition (Partition regularity). Let \mathbf{x} = (x_1, \ldots, x_n), and P(\mathbf{x}) \in \mathbb{Z}[\mathbf{x}] be a polynomial. The equation P(\mathbf{x}) = 0 is partition regular if, given any finite partition of \mathbb{N} = C_1 \sqcup \cdots \sqcup C_k, there exists i \in \{ 1, \ldots, k \} and \mathbf{m} = (m_1, \ldots, m_n) \in C_i^n such that P(\mathbf{m}) = 0.
Clearly, our project had its roots in Ramsey theory and combinatorics. However, our main weapon was actually the application of nonstandard analysis to the problem, in the vein of previous work done by Mauro Di Nasso and Lorenzo Luperi Baglini, among others. First, we must present some definitions.
Ultrafilters generated by hypernaturals. There exists a canonical map {}^* \mathbb{N} \rightarrow \beta \mathbb{N} defined by:
\alpha \mapsto \mathcal{U}_\alpha = \{ A \subseteq \mathbb{N} \mid \alpha \in {}^*\! A \}
Definition (u-equivalence). Fix \zeta, \xi \in {}^* \mathbb{N}. We say \zeta and \xi are u-equivalent, sometimes denoted \zeta \sim \xi, if they generate the same ultrafilter \mathcal{U}_\zeta = \mathcal{U}_\xi \in \beta \mathbb{N}.
The following theorem allowed us to apply nonstandard analysis to the partition regularity problem:
Theorem (Di Nasso & Luperi Baglini). The equation P(x_1, \ldots, x_n) = 0 is partition regular if and only if there exist u-equivalent \xi_1, \ldots, \xi_n \in {}^* \mathbb{N} such that P(\xi_1, \ldots, \xi_n) = 0.
We also built on work done by Joel Moreira on the partition regularity of polynomial configurations, originally in the area of topological dynamics, but later translated to a combinatorial setting. The following corollary of his main theorem is sufficient to demonstrate the idea:
Theorem (Moreira). Let f_1, \ldots, f_n : \mathbb{N} \rightarrow \mathbb{N} be polynomial functions such that either f_1(0) = \cdots = f_n(0) = 0, or f_1(1) = \cdots = f_n(1) = 0. Then the configuration \{ x, xy, x + f_1(y), \cdots, x + f_n(y) \} is partition regular, i.e. for any finite partition of \mathbb{N} = C_1 \sqcup \cdots \sqcup C_k, there exists i \in \{ 1, \ldots, k \} and x,y \in \mathbb{N} such that
\{ x, xy, x + f_1(y), \ldots, x + f_n(y) \} \subseteq C_i.
We translated results about the partition regularity of polynomial configurations into ones about equations by parametrising solutions. For example, for any c \in \mathbb{Z}, the equation P(x,y,z) = x^2 - xy + cz = 0 is partition regular, because it can be parametrised by \{ x,y,z \} = \{ r, r+cs, rs \}, which is a configuration of the form above.
The final product of the project was a paper written by myself, Martino and Joel, published in the European Journal of Combinatorics, vol. 94, no. 103277.