TY - JOUR

T1 - Maximal consistent families of triples

AU - Spencer, Joel

N1 - Funding Information:
* This research is sponsored by the United States Air Force under Project RAND--Contract No. AF 49(638)-1700--monitored by the Directorate of Operational Requirements and Development Plans, Deputy Chief of Staff, Research and Development, Hq USAF. Views or conclusions contained in this Memorandum should not be interpreted as representing the official opinion or policy of the United States Air Force.

PY - 1968/7

Y1 - 1968/7

N2 - The general problem of dividing a set into a maximal collection of subsets in such a way that the subsets will overlap in certain specified ways is a fundamental problem in experimental design and in the design of systems involving shared use of system elements. This memorandum solves this problem in the case for which the subsets are each required to have three elements, and any pair of subsets have at most one element in common. A family F of three element subsets of an n-element set Sn is called n-consistent if the intersection of any two sets of F contain at most one element of Sn. We find maximal (in number of elements) F for all n. For certain n the F are Steiner Triples Systems. The construction of the F is constructive. Structure Theorems are given determining the graph of doublets not covered by triplets in F.

AB - The general problem of dividing a set into a maximal collection of subsets in such a way that the subsets will overlap in certain specified ways is a fundamental problem in experimental design and in the design of systems involving shared use of system elements. This memorandum solves this problem in the case for which the subsets are each required to have three elements, and any pair of subsets have at most one element in common. A family F of three element subsets of an n-element set Sn is called n-consistent if the intersection of any two sets of F contain at most one element of Sn. We find maximal (in number of elements) F for all n. For certain n the F are Steiner Triples Systems. The construction of the F is constructive. Structure Theorems are given determining the graph of doublets not covered by triplets in F.

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U2 - 10.1016/S0021-9800(68)80023-7

DO - 10.1016/S0021-9800(68)80023-7

M3 - Article

AN - SCOPUS:0039313605

SN - 0021-9800

VL - 5

SP - 1

EP - 8

JO - Journal of Combinatorial Theory

JF - Journal of Combinatorial Theory

IS - 1

ER -