Sets of unit fractions without two members whose average is a unit fraction

Abstract

We show that there is a constant $c>0$ such that, for all sufficiently large $N$, there is a subset $A \subseteq \{1,\dots,N\}$ of size $>cN$ such that for any two distinct elements $a,b$ in $A$, the average of $\frac{1}{a}$ and $\frac{1}{b}$ 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.

Disclosure

“nvenient to give a brief narrative of the source of the ideas: The story begins with a calculation of Stijn Cambie [1], who found the largest set A ⊆ {1, . . . , 500} such that for a, b ∈ A with a ̸= b we have a + b ∤ 2ab. The author asked ChatGPT to look for patterns in this set that could give a clue for how to generalize this construction, and it observed that for pairs a, b with a + b | 2ab, the larger one is usually not in A, unless the smaller one”

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2026-08-23T18:56:34.075768Z

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