[Paper Review] Zero-sum Generalized Schur Numbers

Aaron Robertson|arXiv (Cornell University)|Feb 9, 2018

Advanced Combinatorial Mathematics10 references3 citations

TL;DR

This paper introduces zero-sum generalized Schur numbers $ S_{\mathfrak{z}}(k;r) $, defined as the smallest integer $ n $ such that any $ r $-coloring of $[1,n]$ contains a solution to $ \sum_{i=1}^{k-1}x_i = x_k $ with colors summing to $ 0 \pmod{r} $. The author establishes formulas, lower bounds, and exact values for specific $ k $ and $ r $, particularly showing $ S_{\mathfrak{z}}(k;3) = 3k-3 $ for $ k \geq 6 $, and proves $ S_{\mathfrak{z}}(k;k) \geq 2(k^2 - k - 1) $, extending classical Schur theory into zero-sum Ramsey-type settings.

ABSTRACT

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $S_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer $n$ such that every coloring $χ:[1,n] ightarrow \{0,1,\dots,r-1\}$ admits a solution to $\sum_{i=1}^{k-1} x_i = x_k$ with $\sum_{i=1}^{k} χ(x_i) \equiv 0 \,(\mathrm{mod }\,r)$. We give some formulas and lower bounds for various instances.

Motivation & Objective

• To define and study zero-sum variants of generalized Schur numbers, where solutions to $ \sum_{i=1}^{k-1}x_i = x_k $ must have color sums congruent to 0 modulo $ r $.
• To determine exact values and lower bounds for $ S_{\mathfrak{z}}(k;r) $, especially when $ r \mid k $, extending classical Schur and Ramsey theory.
• To investigate the existence and structure of $ r $-zero-sum solutions under constrained colorings, particularly for small $ k $ and $ r $.
• To explore the relationship between zero-sum constraints and rigid arithmetic structures, such as solutions to $ \sum_{i=1}^{k-1}x_i = x_k $.
• To pose open questions on the asymptotic behavior and exact values of $ S_{\mathfrak{z}}(k;r) $, particularly for $ r = k $.

Proposed method

• The paper defines $ S_{\mathfrak{z}}(k;r) $ as the minimal $ n $ such that every $ r $-coloring of $[1,n]$ contains an $ r $-zero-sum solution to $ \sum_{i=1}^{k-1}x_i = x_k $.
• It employs a recursive backtracking algorithm implemented in Fortran (ZSGS.f and ZSGS2.f) to compute $ S_{\mathfrak{z}}(k;r) $ and $ S_{\mathfrak{z,2}}(k;r) $ for small $ k $ and $ r $.
• A custom subroutine efficiently enumerates solutions to $ \sum_{i=1}^{k-1}x_i = t $ under the constraint $ x_1 \leq \cdots \leq x_{k-1} $, reducing computational complexity.
• The author uses modular arithmetic and case analysis modulo $ r $, particularly for $ r = 4 $, to prove nonexistence of zero-sum solutions under certain colorings.
• Structural arguments based on parity and modular constraints are used to show that certain colorings avoid $ r $-zero-sum solutions, yielding lower bounds.
• The paper proves $ S_{\mathfrak{z}}(k;k) \geq 2(k^2 - k - 1) $ via a specific $ k $-coloring that avoids $ k $-zero-sum solutions due to parity mismatches.

Experimental results

Research questions

• RQ1 Is $ S_{\mathfrak{z}}(k;3) = 3k - 3 $ true for all $ k \geq 6 $?
• RQ2 Does $ S_{\mathfrak{z}}(k;4) = 4k - 5 $ hold for all $ k \geq 8 $?
• RQ3 What is the exact value of $ S_{\mathfrak{z,2}}(k;4) $, the two-color zero-sum generalized Schur number for modulus 4?
• RQ4 Is $ S_{\mathfrak{z}}(k;k) $ asymptotically of order $ k^2 $?
• RQ5 Can zero-sum Rado numbers be characterized beyond the results in [10], particularly for structured sequences?

Key findings

• For $ k \geq 6 $, the paper provides evidence that $ S_{\mathfrak{z}}(k;3) = 3k - 3 $, based on computational data and structural analysis.
• The paper proves that $ S_{\mathfrak{z}}(k;4) \geq 4k - 5 $ for $ k \geq 8 $, though the exact value remains open.
• It establishes $ S_{\mathfrak{z}}(k;k) \geq 2(k^2 - k - 1) $, showing that the zero-sum generalized Schur number grows at least quadratically in $ k $.
• For $ r = 2 $, $ S_{\mathfrak{z,2}}(k;k) = k^2 - k - 1 $, matching the classical Schur number due to monochromatic necessity.
• The paper computes exact values for $ S_{\mathfrak{z}}(k;r) $ for small $ k $ and $ r $, including $ S_{\mathfrak{z}}(4;4) = 13 $, $ S_{\mathfrak{z}}(6;3) = 15 $, and $ S_{\mathfrak{z}}(10;4) \geq 45 $.
• It proves that $ S_{\mathfrak{z}}(k;r) = \infty $ when $ r \nmid k $, as monochromatic colorings with color 1 can avoid $ r $-zero-sum solutions.