# Finite and infinite arithmetic progressions in sumsets

### Abstract

We prove that if $A$ is a subset of at least $cn^{1/2}$ elements of $\{1, \dots, n\}$, where $c$ is a sufficiently large constant, then the collection of subset sums of $A$ contains an arithmetic progression of length $n$. As an application, we confirm a long standing conjecture of Erdős and Folkman on complete sequences.

## Authors

Endre Szemerédi

Computer Science Department, Rutgers University, Piscataway, NJ 08854, United States

Van H. Vu

Department of Mathematics, UCSD, La Jolla, CA 92093<, United States