New Proofs for the Disjunctive Rado Number of the Equations $x_1 - x_2 = a$ and $x_1 - x_2 = b$
Published in Graphs and Combinatorics, 2022
Recommended citation: A. Dileep, Jai Moondra, Amitabha Tripathi. (2022). "New Proofs for the Disjunctive Rado Number of the Equations $x_1 - x_2 = a$ and $x_1 - x_2 = b$." Graphs and Combinatorics 38:38. https://link.springer.com/article/10.1007/s00373-021-02400-y
Abstract: Let $m, a, b$ be positive integers, with $\gcd(a,b)=1$. The disjunctive Rado number for the pair of equations $y−x=ma, y−x=mb$, is the least positive integer $R=R_d(ma,mb)$, if it exists, such that every $2$-coloring $\chi$ of the integers in ${ 1,\ldots,R }$ admits a solution to at least one of $\chi(x)=\chi(x+ma), \chi(x)=\chi(x+mb)$. We show that $R_d(ma,mb)$ exists if and only if $ab$ is even, and that it equals $m(a+b−1)+1$ in this case. We also show that there are exactly $2m$ valid $2$-colorings of $[1,m(a+b−1)]$ for the equations $y−x=ma$ and $y−x=mb$, and use this to obtain another proof of the formula for $R_d(ma,mb)$.