On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences

Abstract

We show the existence of a constant $c>0$ such that, for all positive integers $n$, there exist integers $1\le a_1<\cdots <a_k\le n$ such that there are at least $c{n}^2$ distinct integers of the form ${\sum }_{i=u}^v{a}_i$ with $1\le u\le v\le k$. This answers a question of Erdős and Graham. We also prove a non-trivial upper bound on the maximum number of distinct integers of this form and address several open problems.