  We 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 Erd\"os and Szemer\'edi
on sum and product sets.
