Rado's criterion over squares and higher powers (Q2031686)

Schur's theorem from 1916 states that in any finite colouring of the positive integers, there exists a monochromatic solution to the equation \(x + y = z\). Erdős and Graham conjectured that the same is true for the Pythagorean equation \(x^2+y^2=z^2\). This questions is notoriously difficult but the authors manage to obtain a corresponding result with five variables, i.e. their Theorem 1.1 states that in any finite colouring of the positive integers, there exists a monochromatic solution to \(x_1^2 + x_2^2 + x_3^2 + x_4^2 = x_5^2\). This is a consequence of a much more general result concerning partition regularity of polynomials. Given a polynomial \(P \in \mathbb{Z}[x_1, \dotsc, x_s]\) and a set \(S\) we call the equation \(P(x) = 0\) non-trivially partition regular if, in any finite colouring of \(S\), there exists a monochromatic solution \(x \in S^s\) with each variable distinct. Rado's criterion tells that, for \(s \geq 3\) and non-zero integers \(c_1, \dotsc, c_s\), the equation \(\sum_{i= 1}^s c_i x_i = 0\) is non-trivially partition regular over positive integers if and only if there exists a non-empty subset \(I \subseteq \{1, \dotsc, s\}\) such that \(\sum_{i \in I} c_i = 0\). The authors make a vast generalization of this to \(k\)th powers, once the number of variables is sufficiently large. Theorem 1.3 states that, for each \(k \geq 2\), there exists \(s_0(k) \in \mathbb{N}\) such that for \(s \geq s_0(k)\) and non-zero integers \(c_1, \dotsc, c_s\), the equation \[ \sum_{i=1}^s c_i x_i^k = 0 \] is non-trivially partition regular over the positive integers if and only if there exists a non-empty subset \(I \subseteq \{1, \dotsc, s\}\) such that \(\sum_{i \in I} c_i = 0\). Furthermore one can take \(s_0(2) = 5, s_0(3) = 8\) and \[ s_0(k) = k(\log k + \log \log k +2 + O(\log \log k/\log k)). \] The authors also provide results concerning partition-regularity of linear equations over shifted squares and smooth numbers. The proof strategy is explained well in Section 2 of the paper -- the authors need to use variety of methods such as the transference principle and a density increment argument.