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Blocking Sets inSQS(2v)

Published online by Cambridge University Press: 12 September 2008

Mario Gionfriddo
Affiliation:
Dipartimento di Matematica, Città Universitaria, Viale A, Doria 6, 95125 Catania, Italy.
Salvatore Milici
Affiliation:
Dipartimento di Matematica, Città Universitaria, Viale A, Doria 6, 95125 Catania, Italy.
Zsolt Tuza
Affiliation:
Computer and Automation Institute, Hungarian Academy of Sciences, H-llll Budapest, Kende u. 13–17, Hungary

Abstract

A Steiner quadruple systemSQS(v) of orderv is a family ℬ of 4-element subsets of av-element setV such that each 3-element subset ofV is contained in precisely oneB ∈ℬ. We prove that ifT ∩B ≠ ø for allB ∈ℬ (i.e., ifT is atransversal), then |T| ≥v/2, and ifT is a transversal of cardinality exactlyv/2, thenV \ T is a transversal as well (i.e.,T is ablocking set). Also, in respect of the so-called ‘doubling construction’ that producesSQS(2v) from two copies ofSQS(v), we give a necessary and sufficient condition for this operation to yield a Steiner quadruple system with blocking sets.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1994

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