2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68459We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ... \cdot A| \geq |A|^b$$ where $A + ... + A$ and $A \cdot ... \cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erdös and Szemerédi on sum and product sets.33 pages, no figures, submitted, J. Amer. Math. Soc. (Proxy submission)CombinatoricsNumber Theory11P70text