• Source: Sum-free set

In additive combinatorics and number theory, a subset A of an abelian group G is said to be sum-free if the sumset A + A is disjoint from A. In other words, A is sum-free if the equation



a
+
b
=
c


{\displaystyle a+b=c}

has no solution with



a
,
b
,
c
∈
A


{\displaystyle a,b,c\in A}

.
For example, the set of odd numbers is a sum-free subset of the integers, and the set {N + 1, ..., 2N } forms a large sum-free subset of the set {1, ..., 2N }. Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero nth powers of the integers is a sum-free set.
Some basic questions that have been asked about sum-free sets are:

How many sum-free subsets of {1, ..., N } are there, for an integer N? Ben Green has shown that the answer is



O
(

2

N

/

2


)


{\displaystyle O(2^{N/2})}

, as predicted by the Cameron–Erdős conjecture.
How many sum-free sets does an abelian group G contain?
What is the size of the largest sum-free set that an abelian group G contains?
A sum-free set is said to be maximal if it is not a proper subset of another sum-free set.
Let



f
:
[
1
,
∞
)
→
[
1
,
∞
)


{\displaystyle f:[1,\infty )\to [1,\infty )}

be defined by



f
(
n
)


{\displaystyle f(n)}

is the largest number



k


{\displaystyle k}

such that any subset of



[
1
,
∞
)


{\displaystyle [1,\infty )}

with size n has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma,




lim

n





f
(
n
)

n




{\displaystyle \lim _{n}{\frac {f(n)}{n}}}

exists. Erdős proved that




lim

n





f
(
n
)

n


≥


1
3




{\displaystyle \lim _{n}{\frac {f(n)}{n}}\geq {\frac {1}{3}}}

, and conjectured that equality holds. This was proved by Eberhard, Green, and Manners.




See also


Erdős–Szemerédi theorem
Sum-free sequence




References

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