On the minimum number of monochromatic solutions to the strict Schur inequality in 2-colored integer intervals with negative left endpoint

Abstract

Kosek, Robertson, Sabo, and Schaal studied the minimum number \(M_k(n)\) of monochromatic solutions to the strict Schur inequality system $x_1\le x_2\le x_3$ and $x_1+x_2<x_3$ in \(2\)-colorings of \([k+1,k+n]\). They proved that for every fixed \(k\ge 0\), $M_k(n)=\frac{n^3}{12(1+2\sqrt{2}{)}^2}(1+o_k(1)),$ and left open the case \(k\le -2\). In this paper, we resolve that remaining range.