What could be simpler than to study sums and products of integers? Well maybe it is not so simple since there is a major unsolved problem: for arbitrarily large numbers N, can there be sets of N positive integers where both the number of pairwise sums and pairwise product is less than N^{3/2}?

No one knows. This talk is directed at another problem concerning sums and products, namely how dense can a set of positive integers be if it contains none of its pairwise sums and products? For example, take the numbers that are 2 or 3 more than a multiple of 5, a set with density 2/5. Can you do better?