On two conjectures for generalized off-diagonal Schur numbers

Abstract

For an integer $t\ge 3$, 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 $k\ge 1$ and $t_0,t_1,\dots ,t_{k-1}\ge 3$, 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: $S(3;3,t,u)>3tu-tu-u-1$ for $3=t<u$ and $3<t\le u$. In this paper, we confirm this conjecture. At the same paper, they also conjecture that $S(3;s,t,u)=stu-tu-u-1$ for $4\le s\le t\le u$. 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 $u$.

13 pages