  A linear equation is $r$-regular, if, for every $r$-coloring of the positive
integers, there exist positive integers of the same color which satisfy the
equation. In 2005, Fox and Radoicic conjectured that the equation $x_1 + 2x_2 +
\cdots + 2^{n-2}x_{n-1} - 2^{n-1}x_n = 0$, for any $n \geq 2$, has a degree of
regularity of $n-1$, which would verify a conjecture of Rado from 1933. Rado's
conjecture has since been verified with a different family of equations. In
this paper, we show that Fox and Radoicic's family of equations indeed have a
degree of regularity of $n-1$. We also provide a few extensions of this result.

Combinatorics