On Rado numbers for equations with unit fractions

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 $f_3(k)=O(k^{43})$.