Erdős–Szemerédi theorem

In arithmetic combinatorics, the Erdős–Szemerédi theorem, proven by Paul Erdős and Endre Szemerédi in 1983, states that, for every finite set of real numbers, either the pairwise sums or the pairwise products of the numbers in the set form a significantly larger set. More precisely, it asserts the existence of positive constants c and ${\displaystyle \varepsilon }$ such that

whenever A is a finite non-empty set of real numbers of cardinality |A|, where ${\displaystyle A+A=\{a+b:a,b\in A\}}$ is the sum-set of A with itself, and ${\displaystyle A\cdot A=\{ab:a,b\in A\}}$.

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