Upper and Lower Bounds on Zero-Sum Generalized Schur Numbers

TL;DR

This paper establishes bounds and exact values for zero-sum generalized Schur numbers, revealing their behavior depending on parameters like the number of colors and the size of the sequence, especially for prime moduli.

Contribution

The authors derive tight bounds and exact values for zero-sum generalized Schur numbers, extending understanding of their properties in combinatorial number theory.

Findings

For k > r, bounds are kr - r - 1 ≤ S_z(k,r) ≤ kr - 1.

When r is an odd prime, S_z(k,r) equals kr - r.

The exact value of S_z(k,r) is determined in specific cases.

Abstract

Let $S_{\mathfrak{z}}(k,r)$ be the least positive integer such that for any $r$-coloring $\chi : \{1,2,\dots,S_{\mathfrak{z}}(k,r)\} \longrightarrow \{1, 2, \dots, r\}$, there is a sequence $x_1, x_2, \dots, x_k$ such that $\sum_{i=1}^{k-1} x_i = x_k$, and $\sum_{i=1}^{k} \chi(x_i) \equiv 0 \pmod{r}$. We show that when $k$ is greater than $r$, $kr - r - 1 \le S_{\mathfrak{z}}(k,r) \le kr - 1$, and when $r$ is an odd prime, $S_{\mathfrak{z}}(k,r)$ is in fact equal to $kr - r$.