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 + ⋯ + 1 / x k = 1 / y , where x 1 , ... , x k are not necessarily distinct. Brown and Rödl [3] 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 ⊆ N such that every 2-coloring of A has a monochromatic solution to the linear equation x 1 + ⋯ + 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 et al. [2] implies that f 3 (k) = O (k 43). [ABSTRACT FROM AUTHOR]