Yang Gang (Yokohama): Complete solution to a conjecture of Butler, Costello, and Graham

In 2010, Butler, Costello, and Graham proposed a conjecture: Let $ax + by = az$ be an equation, where $a, b$ are integers.

Denote by $R,B$ the colors red and blue, respectively. 

$(i)$ If $b>a\geq 2$ and $\gcd(a, b)=1$, then the coloring that gives the minimum number of monochromatic solutions over any $2$-coloring of $[1, n]$ is $[(R^{a-1}, B)^{\frac{n}{b}},R^{(\frac{b-a}{b})n}]$.

$(ii)$ If $a>b\geq 2$ and $\gcd(a, b)=1$, then

the coloring that gives the minimum number of monochromatic solutions over

any $2$-coloring of $[1, n]$ is $[(R^{a-1}, B)^{\frac{n}{a}}]$. In this paper, we confirm this conjecture.