On two conjectures for generalized off-diagonal Schur numbers

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 $u$.