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

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2: KRSS, Theorem 1
• Theorem 1.3: KRSS Theorem 7
• Theorem 1.4
• Proposition 2.1
• proof
• Theorem 3.1
• proof
• Lemma 4.1
• proof