Sum-free set explained
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
has no solution with
.
For example, the set of odd numbers is a sum-free subset of the integers, and the set forms a large sum-free subset of the set . 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 are there, for an integer N? Ben Green has shown[1] that the answer is
, as predicted by the
Cameron–Erdős conjecture.
[2] - How many sum-free sets does an abelian group G contain?[3]
- 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
be defined by
is the largest number
such that any subset of
with size
n has a sum-free subset of size
k. The
function is
subadditive, and by the Fekete subadditivity lemma,
exists.
Erdős proved that
, and
conjectured that equality holds.
[4] This was proved by Eberhard, Green, and Manners.
[5] See also
Notes and References
- Ben . Green . math.NT/0304058 . The Cameron–Erdős conjecture . . 36 . 6 . November 2004 . 769–778 . 10.1112/S0024609304003650 . 2083752.
- P.J. Cameron and P. Erdős, "On the number of sets of integers with various properties", Number Theory (Banff, 1988), de Gruyter, Berlin 1990, pp. 61-79; see Sloane
- Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005.
- P. Erdős, "Extremal problems in number theory", Matematika, 11:2 (1967), 98–105; Proc. Sympos. Pure Math., Vol. VIII, 1965, 181–189
- Eberhard . Sean . Green . Ben . Manners . Freddie . 2014 . Sets of integers with no large sum-free subset . Annals of Mathematics . 180 . 2 . 621–652 . 0003-486X.
