On a Rado type problem for homogeneous second order linear recurrences (Q2380472)

Summary: We introduce a Ramsey type function \(S(r; a,b,c)\) as the maximum \(s\) such that for any \(r\)-coloring of \(\mathbb{N}\) there is a monochromatic sequence \(x_1,x_2,\dots, x_s\) satisfying a homogeneous second-order linear recurrence \(ax_i+ bx_{i+1}+ cx_{i+2}= 0\), \(1\leq i\leq s- 2\). We investigate \(S(2; a,b,c)\) and evaluate its values for a wide class of triples \((a,b,c)\).