Sum-Free Set
المؤلف:
Abbott, H. L. and Moser, L.
المصدر:
"Sum-Free Sets of Integers." Acta Arith. 11
الجزء والصفحة:
...
5-11-2020
799
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 ![<span style=]()
{1,2,3}" src="https://mathworld.wolfram.com/images/equations/Sum-FreeSet/Inline4.gif" style="height:15px; width:47px" /> are 
, ![<span style=]()
{1}" src="https://mathworld.wolfram.com/images/equations/Sum-FreeSet/Inline6.gif" style="height:15px; width:17px" />, ![<span style=]()
{2}" src="https://mathworld.wolfram.com/images/equations/Sum-FreeSet/Inline7.gif" style="height:15px; width:17px" />, ![<span style=]()
{3}" src="https://mathworld.wolfram.com/images/equations/Sum-FreeSet/Inline8.gif" style="height:15px; width:17px" />, ![<span style=]()
{1,3}" src="https://mathworld.wolfram.com/images/equations/Sum-FreeSet/Inline9.gif" style="height:15px; width:32px" />, and ![<span style=]()
{2,3}" src="https://mathworld.wolfram.com/images/equations/Sum-FreeSet/Inline10.gif" style="height:15px; width:32px" />. The numbers of sum-free subsets of ![<span style=]()
{1,2,...,n}" src="https://mathworld.wolfram.com/images/equations/Sum-FreeSet/Inline11.gif" style="height:15px; width:69px" /> 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.
الاكثر قراءة في نظرية الاعداد
اخر الاخبار
اخبار العتبة العباسية المقدسة
