---
tag: YP9194YVVN
kind: statement
role: proposition
name: Erdős–Szemerédi theorem
aliases: [sum-product-phenomenon, sum-product-theorem]
sources:
  - wikipedia: en:35021032@r1374258512
    locator: Erdős–Szemerédi theorem
    note: adopted
    mention: wikipedia://en/35021032
    sections: [Statement]
same_as:
  wikidata: Q3526986
  wikipedia: Erdős–Szemerédi theorem
inventory: [T60ZGJDZT5]
---

## Statement

In arithmetic combinatorics, the **Erdős–Szemerédi theorem** states that for every [[finite set|070R982G4B]] $A$ of [[integers|0C4JS9RF5M]], 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 $A\subset\mathbb{N}$,