Erdős–Szemerédi theorem

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Published on August 25, 2021 6:52:38 PM ● Video Link: https://www.youtube.com/watch?v=vrnNfwKXVG4

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The Erdős–Szemerédi theorem in arithmetic combinatorics states that for every finite set



A


{\displaystyle A}
of integers, at least one of



A
+
A


{\displaystyle A+A}
, the set of pairwise sums or



A
⋅
A


{\displaystyle A\cdot A}
, the set of pairwise products form a significantly larger set. More precisely, the Erdős–Szemerédi theorem states that there exist positive constants c and



ε


{\displaystyle \varepsilon }
such that for any non-empty set



A
⊂

N



{\displaystyle A\subset \mathbb {N} }





max
(

|

A
+
A

|

,

|

A
⋅
A

|

)
≥
c

|

A


|


1
+
ε




{\displaystyle \max(|A+A|,|A\cdot A|)\geq c|A|^{1+\varepsilon }}
.It was proved by Paul Erdős and Endre Szemerédi in 1983. The notation




|

A

|



{\displaystyle |A|}
denotes the cardinality of the set



A


{\displaystyle A}
.
The set of pairwise sums is



A
+
A
=
{
a
+
b
:
a
,
b
∈
A
}


{\displaystyle A+A=\{a+b:a,b\in A\}}
and is called sum set of



A


{\displaystyle A}
.
The set of pairwise products is



A
⋅
A
=
{
a
b
:
a
,
b
∈
A
}


{\displaystyle A\cdot A=\{ab:a,b\in A\}}
and is called the product set of



A


{\displaystyle A}
.
The theorem is a version of the maxim that additive structure and multiplicative structure cannot coexist. It can also be viewed as an assertion that the real line does not contain any set resembling a finite subring or finite subfield; it is the first example of what is now known as the sum-product phenomenon, which is now known to hold in a wide variety of rings and fields, including finite fields.

Source: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szemer%C3%A9di_theorem
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