Indecomposable triple systems |
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Authors: | Earl S. Kramer |
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Affiliation: | University of Nebraska, Lincoln, Neb. 68508, USA |
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Abstract: | A t-design (λ, t, d, n) is a system of sets of size d from an n-set S, such that each t subset of S is contained in exactly λ elements of . A t-design is indecomposable (written IND(λ, t, d, n)) if there does not exist a subset ? such that is a (λ, t, d, n) for some λ, 1 ? λ < λ. A triple system is a (λ; 2, 3, n). Recursive and constructive methods (several due to Hanani) are employed to show that: (1) an IND(2; 2, 3, n) exists for n ≡ 0, 1 (mod 3), n ? 4 and n ≡ 7 (designs of Bhattacharya are used here), (2) an IND(3; 2, 3, n) exists for n odd, n ? 5, (3) if an IND(λ, 2, 3, n) exists, n odd, then there exists an infinite number of indecomposable triple systems with that λ. |
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