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Steiner triple systems with disjoint or intersecting subsystems
Author(s) -
Colbourn Charles J.,
Oravas Monica A.,
Rees Rolf S.
Publication year - 2000
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/(sici)1520-6610(2000)8:1<58::aid-jcd8>3.0.co;2-2
Subject(s) - disjoint sets , steiner system , mathematics , combinatorics , order (exchange) , discrete mathematics , economics , finance
Abstract The existence of incomplete Steiner triple systems of order υ having holes of orders w and u meeting in z elements is examined, with emphasis on the disjoint ( z = 0) and intersecting ( z = 1) cases. When $w \ge u$ and $\upsilon=2w+u-2z$ , the elementary necessary conditions are shown to be sufficient for all values of z . Then for $z\in\{0,1\}$ and υ “near” the minimum of $2w+u-2\,z$ , the conditions are again shown to be sufficient. Consequences for larger orders are also discussed, in particular the proof that when one hole is at least three times as large as the other, the conditions are again sufficient. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 58–77, 2000