  Let $f_r(k)$ be the smallest positive integer $n$ such that every
$r$-coloring of $\{1,2,...,n\}$ has a monochromatic solution to the nonlinear
equation \[1/x_1+\cdots+1/x_k=1/y,\] where $x_1,...,x_k$ are not necessarily
distinct. Brown and R\"{o}dl [Bull. Aust. Math. Soc. 43(1991): 387-392] proved
that $f_2(k)=O(k^6)$. In this paper, we prove that $f_2(k)=O(k^3)$. The main
ingredient in our proof is a finite set $A\subseteq\mathbb{N}$ such that every
$2$-coloring of $A$ has a monochromatic solution to the linear equation
$x_1+\cdots+x_k=y$ and the least common multiple of $A$ is sufficiently small.
This approach can also be used to study $f_r(k)$ with $r>2$. For example, a
recent result of Boza, Mar\'{i}n, Revuelta, and Sanz [Discrete Appl. Math.
263(2019): 59-68] implies that $f_3(k)=O(k^{43})$.
