The Cameron-Erdős conjecture asserts that the number of sum-free subsets of
is
(Cameron and Erdős 1990). Here a sum-free set
contains no solution to with , , and in the set, including solutions with
,
and
denotes big-O notation.
For ,
1, ..., the counts are 1, 2, 3, 6, 9, 16, 24, 42, 61, ... (OEIS A007865),
including the empty set.
Green (2004) and Sapozhenko (2008) proved the conjecture independently. Every subset of is a sum-free
set, giving . Thus the proved bound determines up to absolute constant factors. Here and denote the floor function
and ceiling function.
Bloom, T. F. "Erdős Problem 748." Accessed Oct. 26, 2026. https://www.erdosproblems.com/748.Cameron,
P. J. and Erdős, P. "On the Number of Sets of Integers with Various
Properties." In Number Theory: Proceedings of the First Conference of the
Canadian Number Theory Association Held in Banff, Alberta, April 17-27, 1988
(Ed. R. A. Mollin). Berlin, Germany: de Gruyter, pp. 61-79, 1990.Green,
B. "The Cameron-Erdős Conjecture." Bull. London Math. Soc.36,
769-778, 2004. https://doi.org/10.1112/S0024609304003650.Sapozhenko,
A. A. "The Cameron-Erdős Conjecture." Disc. Math.308,
4361-4369, 2008. https://doi.org/10.1016/j.disc.2007.08.103.Sloane,
N. J. A. Sequence A007865in "The
On-Line Encyclopedia of Integer Sequences."
of sum-free subsets of
is 
(Cameron and Erdős 1990). Here a sum-free set
contains no solution to 
with 
, 
, and 
in the set, including solutions with
,
and 
denotes big-O notation.
,
1, ..., the counts are 1, 2, 3, 6, 9, 16, 24, 42, 61, ... (OEIS A007865),
including the empty set.
is a sum-free
set, giving ![f(n)>=2^([n/2])](/images/equations/Cameron-ErdosConjecture/Inline12.svg)
. Thus the proved bound determines 
up to absolute constant factors. Here 
and ![[x]](/images/equations/Cameron-ErdosConjecture/Inline15.svg)
denote the floor function
and ceiling function.
