Abstract: We walk around the seemingly innocent result, that if A and B are finite sets of integers (or vectors or elements of a torsion free commutative group), G is a "large" subset of pairs (a,b) from A\times B, but the sums a+b:(a,b)\in G define only a few different elements, then there are "large" subsets A'\subset A, B'\subset B such that ALL of their sums a+b:a\in A', b\in B' define only a few different elements.
The discussion of several applications will show the importance of this result. We also indicate the main steps of a simple, elegant, completely elementary proof. No special prerequisite is needed.
This talk will be accessible to undergraduates.