On two conjectures for generalized off-diagonal Schur numbers

TL;DR

This paper confirms a conjecture about the lower bounds of generalized off-diagonal Schur numbers for specific parameters and provides recursive lower bounds and upper bounds for these numbers in general.

Contribution

It verifies a specific conjecture on Schur numbers and introduces recursive lower bounds and upper bounds for generalized Schur numbers.

Findings

Confirmed Ahmed and Schaal's conjecture for certain parameters.

Derived recursive lower bounds for generalized Schur numbers.

Established upper bounds for large parameters.

Abstract

For an integer $t \geq 3$, let $\mathcal{L}(t)$ denote the linear equation $x_1 + x_2 + \cdots + x_{t-1} = x_t,$ where all variables are positive integers. For integers $k \geq 1$ and $t_0,t_1,\dots,t_{k-1} \geq 3$, the generalized Schur number $S(k;t_0,t_1,\dots,t_{k-1})$ is the least positive integer $N$ such that every $k$-coloring of $[1,N]$, for some $i \in \{0,1,\dots,k-1\}$, a solution to $\mathcal{L}(t_i)$ with all variables monochromatic in color $i$. In 2015, Ahmed and Schaal proposed a conjecture: $S(3 ; 3, t, u)>3 t u-t u-u-1$ for $3=t<u$ and $3<t \leq u$. In this paper, we confirm this conjecture. At the same paper, they also conjecture that $S(3 ; s, t, u)=s t u-t u-u-1$ for $4 \leq s \leq t \leq u$. Motivated by the second conjecture, we give a recursive lower bound of $S(r; k_0, k_1, \dots, k_{r-1})$ and upper bounds for $S(r; k_0, k_1, \dots, k_{r-1})$ and $S(r;k_0,\dots,k_{r-2},u)$ for all sufficiently large $u$.