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 $x_1+x_2&lt;x_3$ in (2)-colorings of ([k+1,k+n]). They proved that for every fixed (k\ge 0), $$M_k(n)= \frac{n<sup>3}{12(1+2\sqrt2)<sup>2}(1+o_k(1)),$$ and left open the case (k\le -2). In this paper, we resolve that remaining range.

• The paper determines exact counts and asymptotic behavior for monochromatic solutions in 2-colored signed integer intervals, proving the universal cubic scaling with a leading constant of approximately 0.00569.
• It employs a discrete optimization framework by partitioning the interval into negative, zero, and positive segments to manage color assignments effectively.
• Numerical results confirm that the inclusion of a fixed negative block does not affect the cubic asymptotic behavior, unifying the treatment of nonnegative and signed intervals.

Minimum Monochromatic Solutions to the Strict Schur Inequality in Signed Intervals

Introduction and Background

The paper addresses the multiplicity problem for the strict Schur inequality system $x_1 \leq x_2 \leq x_3$, $x_1 + x_2 < x_3$ under 2-colorings of signed integer intervals. While prior work by Kosek, Robertson, Sabo, and Schaal determined the minimum number of monochromatic solutions, $M_k(n)$, for integer intervals with nonnegative left endpoint ($k \geq 0$) [5], the case $k \leq -2$—intervals that begin with negative elements—remained unresolved. This work fills that gap by establishing exact counts and asymptotics for the minimum number of monochromatic solutions in this nonpositive setting.

Main Results

The critical advance is the exact reduction of $M_k(n)$ for intervals $[k+1, k+n]$ where $k \leq -2$ (i.e., for $k = -t$, $t \geq 2$). The authors show that, despite the added complexity from the fixed nonpositive block in the interval, the asymptotic behavior remains governed by the same cubic term as in the case $x_1 + x_2 < x_3$0:

$x_1 + x_2 < x_3$1

This result is underpinned by an exact, finite reduction to a discrete optimization problem in the parameters specifying color assignments in the nonpositive segment and a block-parameter describing the coloring of the positive segment. For $x_1 + x_2 < x_3$2, the key is a decomposition into negative, zero, and positive parts, reducing the problem to block colorings on a positive tail after fixing colors on the negative segment and zero.

For small intervals ($x_1 + x_2 < x_3$3), the purely nonpositive regime is addressed, with explicit formulae for $x_1 + x_2 < x_3$4, leveraging combinatorial enumeration of multisets.

Methods

To facilitate exact counts, the paper classifies monochromatic solutions to the system $x_1 + x_2 < x_3$5, $x_1 + x_2 < x_3$6 in four types based on the locations of the variables (in negative, zero, or positive parts of the interval): PPP, QQQ, QQP, QPP. Carefully constructed expressions enumerate solutions of each type, parameterized by the 2-coloring of the negative block and the two-block coloring structure of the positive tail, which (by prior results and symmetry) is always optimal.

The technical basis is a reduction of the problem to optimizing a function $x_1 + x_2 < x_3$7 in discrete parameters $x_1 + x_2 < x_3$8, where $x_1 + x_2 < x_3$9 is the count of negative entries colored 0, $M_k(n)$0 is the color of 0, and $M_k(n)$1 is the count of positive entries colored 0 in the tail. By symmetry, only $M_k(n)$2 need be considered.

The asymptotic regime is reached by observing that, for fixed $M_k(n)$3 (i.e., the negative block size), the leading cubic term in $M_k(n)$4 arises from the positive tail and matches the constant of the nonnegative case, with only lower order corrections from the negative part.

Numerical Results and Claims

The core numerical finding is the explicit constant in the cubic asymptotic term,

$M_k(n)$5

appearing in $M_k(n)$6 for all fixed $M_k(n)$7 (including $M_k(n)$8). This unifies the asymptotic behavior for all shifts of the interval, demonstrating that the presence of an arbitrary (but fixed) negative block does not alter the cubic scaling nor the leading constant.

An explicit, computationally tractable minimization is provided for finite $M_k(n)$9, making the result actionable for both theoretical and computational enumeration.

Implications and Future Directions

This work fully resolves the minimum-multiplicity problem for the strict Schur inequality on 2-colored integer intervals, including the nuanced signed case previously open. The confirmation that the cubic constant is universal for all fixed interval shifts is significant—it suggests a robustness of multiplicity asymptotics to the addition of bounded "obstacles" (here, negative or zero elements) in the interval analyzed.

From a theoretical perspective, the result motivates analogous analyses for other linear systems or for situations where the nonpositive block does not remain fixed, but grows with $k \geq 0$0. Moreover, the paper raises the question of whether similarly precise finite reductions can be realized for other systems of inequalities where fixed substructures interact with Ramsey-type constraints over long positive tails.

In practical terms, the enumeration techniques and reduction machinery developed herein could facilitate advances in computational Ramsey theory for more general equations, inequalities, and coloring rules.

Conclusion

The authors close the previously open case of 2-color strict Schur inequality multiplicity on integer intervals with negative left endpoints. They provide exact finite reductions and show that the leading-order asymptotic constant is invariant under fixed shifts—i.e., unaffected by the addition of a bounded nonpositive block. This technical achievement enriches the structural theory of Ramsey multiplicity for additive inequalities and signals a direction for further study in mixed-sign intervals and more complex combinatorial systems.