A sum-free set subset of an additive group
for which the intersection of sumset
For example, the sum-free sets of A007865 ).
The numbers of sum-free sets can be computed in the Wolfram Language using the following code (P. Abbott, pers. comm., Nov. 24,
2005):
NumbersOfSumFreeSets[nmax_] := Module[{n = 0},
Last[Reap[Nest[(++n; Sow[Length[#]];
Union[#, Union[#, {n}]& /@
Select[#, Intersection[#, n - #] == {}&]])&,
{{}}, nmax + 1]
]
]
]
The Cameron-Erdős conjecture determines the order of growth of the number of sum-free subsets
of
See also A -Sequence,
Cameron's Sum-Free Set Constant ,
Double-Free Set ,
Hofstadter Sequences ,
Prime
Number of Measurement ,
s -Additive Sequence,
Schur Number ,
Schur's
Problem ,
Stöhr Sequence ,
Triple-Free
Set
Explore with Wolfram|Alpha
References Abbott, H. L. and Moser, L. "Sum-Free Sets of Integers." Acta Arith. 11 , 393-396, 1966. https://doi.org/10.4064/aa-11-4-393-396 . 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 Cameron, P. J. and Erdős,
P. "Notes on Sum-Free and Related Sets." Combin. Probab. Comput. 8 ,
95-107, 1999. Exoo, G. "A Lower Bound for Schur Numbers and Multicolor
Ramsey Numbers of Elec. J. Combin. 1 , R8, 1994. https://doi.org/10.37236/1188 . Finch,
S. R. "Cameron's Sum-Free Set Constants." §2.25 in Mathematical
Constants. Fredricksen, H. and Sweet, M. M. "Symmetric Sum-Free
Partitions and Lower Bounds for Schur Numbers." Elec. J. Combin. 7 ,
R32, 2000. https://doi.org/10.37236/1510 . Green,
B. "The Cameron-Erdős Conjecture." Bull. London Math. Soc. 36 ,
769-778, 2004. https://doi.org/10.1112/S0024609304003650 . Sloane,
N. J. A. Sequence A007865 in "The
On-Line Encyclopedia of Integer Sequences." Wallis, W. D.;
Street, A. P.; and Wallis, J. S. Combinatorics:
Room Squares, Sum-Free Sets, Hadamard Matrices. Wang, E. T. H. "On Double-Free Sets of Integers."
Ars Combin. 28 , 97-100, 1989. Referenced on Wolfram|Alpha Sum-Free
Set
Cite this as:
Weisstein, Eric W. "Sum-Free Set." From
MathWorld https://mathworld.wolfram.com/Sum-FreeSet.html
Subject classifications More... Less...
is a subset of an additive group
for which the intersection of 
and the sumset 
is empty. The two summands in 
need not be distinct.
are 
, 
, 
,
, 
, and 
. The numbers of sum-free subsets of 
for 
, 1, ... are 1, 2, 3, 6, 9, 16, 24, 42, 61, ... (OEIS A007865).
.
."
Elec. J. Combin. 1, R8, 1994.