Sum-Free Set

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

For example, the sum-free sets of  are , , , , , and . The numbers of sum-free subsets of  for , 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]
      ]
    ]
  ]

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 ." Electronic J. Combinatorics 1, No. 1, R8, 1-3, 1994. https://www.combinatorics.org/Volume_1/Abstracts/v1i1r8.html.

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." Electronic J. Combinatorics 7, No. 1, R32, 1-9, 2000. https://www.combinatorics.org/Volume_7/Abstracts/v7i1r32.html.

Green, B. "The Cameron-Erdős Conjecture." Apr. 4, 2003. https://www.arxiv.org/abs/math.NT/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.