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Determination of a Subset from Certain Combinatorial Properties

  • David G. Cantor (a1) and W. H. Mills (a1)

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Let N be a finite set of n elements. A collection ﹛S1, S2, … , Sm﹜ of subsets of N is called a determining collection if an arbitrary subset T of N is uniquely determined by the cardinalities of the intersections SiT, 1 ≤ im. The purpose of this paper is to study the minimum value D(n) of m for which a determining collection of m subsets exists.

This problem can be expressed as a coin-weighing problem (1; 7).

In a recent paper Cantor (1) showed that D(n) = O(n/log log n), thus proving a conjecture of N. J. Fine (3) that D(n) = o(n). More recently Erdös and Rényi (2), Söderberg and Shapiro (7), Berlekamp, Mills, and Leo Moser have independently found proofs that D(n) = O(n/log n).

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References

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1. Cantor, David G., Determining a set from the cardinalities of its intersections with other sets, Can. J. Math., 16 (1964), 9497.
2. Erdös, P. and Rényi, A., On two problems of information theory, Publ. Hung. Acad. Sci., 8 (1963), 241254.
3. Fine, N. J., Solution E1399, Amer. Math. Monthly, 67 (1960), 697698.
4. Lindström, B., On a combinatory detection problem, Publ. Hung. Acad. Sci., 9 (1964), 195207.
5. Lindström, B., On a combinatorial problem in number theory, Can. Math. Bull., 4(1965), 477490.
6. Ryser, H. J., Maximal determinants in combinatorial investigations, Can. J. Math., 8 (1956), 245249.
7. Söderberg, Staffan and Shapiro, H. S., A combinatory detection problem, Amer. Math. Monthly, 70 (1963), 10661070.
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