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

TL;DR

This paper determines the minimum number of monochromatic solutions to a specific inequality in 2-colored integer intervals, completing previous results by resolving the case for negative left endpoints.

Contribution

It extends prior work by solving the open case for negative left endpoints in the minimum monochromatic solutions problem.

Findings

Established the exact minimum number of solutions for negative left endpoints.

Confirmed the asymptotic formula for all integer intervals with both positive and negative endpoints.

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 \(k\ge 0\), $M_k(n)= \frac{n^3}{12(1+2\sqrt2)^2}(1+o_k(1)),$ and left open the case \(k\le -2\). In this paper, we resolve that remaining range.