2026/07/16 by Will Sawin
#math.NT #math.CO
We show that there is a constant c>0 such that, for all sufficiently large N, there is a subset A ⊆ \1,…,N\ of size >cN such that for any two distinct elements a,b in A, the average of (1)/(a) and (1)/(b) is not a unit fraction, negatively answering a question of Erdős and Graham. This also gives the best known lower bounds on the maximum size of a set of unit fractions without non-trivial three-term arithmetic progressions.