On Rado numbers for equations with unit fractions

Abstract

Let fr(k) be the smallest positive integer n such that every r-coloring of {1,2,…,n} has a monochromatic solution to the nonlinear equation1/x1+⋯+1/xk=1/y, where x1,…,xk are not necessarily distinct. Brown and Rödl [3] proved that f2(k)=O(k6). In this paper, we prove that f2(k)=O(k3). The main ingredient in our proof is a finite set A⊆N such that every 2-coloring of A has a monochromatic solution to the linear equation x1+⋯+xk=y and the least common multiple of A is sufficiently small. This approach can also be used to study fr(k) with r>2. For example, a recent result of Boza et al. [2] implies that f3(k)=O(k43).