We prove an old conjecture of Erd˝s and Graham on sums of unit fractions: o There exists a constant b 0 such that if we r-color the integers in [2, br ], then there exists a monochromatic set S such that n∈S 1/n = 1. 1. Introduction We will prove a result on unit fractions which has the following corollary. Corollary. There exists a constant b so that for every partition of the integers in [2, br ] into r classes, there is always one class containing a subset S with the property n∈S 1/n = 1
