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Sum-Free Set

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A sum-free set S is a set for which the intersection of S and the sumset S+S is empty.

For example, the sum-free sets of {1,2,3} are emptyset, {1}, {2}, {3}, {1,3}, and {2,3}. The numbers of sum-free subsets of {1,2,...,n} for n=0, 1, ... are 1, 2, 3, 6, 9, 16, 24, 42, 61, ... (OEIS 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]
      ]
    ]
  ]

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

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References

Abbott, H. L. and Moser, L. "Sum-Free Sets of Integers." Acta Arith. 11, 392-396, 1966.Cameron, P. J. and Erdős, P. "On the Number of Sets of Integers with Various Properties." 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: de Gruyter, pp. 61-79, 1990.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 K_3." Elec. J. Combin. 1, No. 1, R8, 1-3, 1994. https://doi.org/10.37236/1188.Finch, S. R. "Cameron's Sum-Free Set Constants." §2.25 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 180-183, 2003.Fredricksen, H. and Sweet, M. M. "Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers." Elec. J. Combin. 7, No. 1, R32, 1-9, 2000. https://doi.org/10.37236/1510.Green, B. "The Cameron-Erdős Conjecture." Apr. 4, 2003. https://arxiv.org/abs/math/0304058.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. New York: Springer-Verlag, 1972.Wang, E. T. H. "On Double-Free Sets of Integers." Ars Combin. 28, 97-100, 1989.

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Sum-Free Set

Cite this as:

Weisstein, Eric W. "Sum-Free Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Sum-FreeSet.html

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