Erdős–Szemerédi theorem

Statement

In arithmetic combinatorics, the Erdős–Szemerédi theorem states that for every finite set $A$ of integers, at least one of the sets $A+A$ and $A\cdot A$ (the sets of pairwise sums and pairwise products, respectively) form a significantly larger set. More precisely, the Erdős–Szemerédi theorem states that there exist positive constants $c$ and $\varepsilon$ such that, for any non-empty set $A\subset\mathbb{N}$,