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On the Size of a Maximum Transversal in a Steiner Triple System

Published online by Cambridge University Press:  20 November 2018

A. E. Brouwer
A. E. Brouwer*
Affiliation:
Stichting Mathematisch Centrum, Amsterdam, The Netherlands
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Let (X, ) be a Steiner triple system on v = |X| points, and suppose that is a partial parallel class (transversal, clear set, set of pairwise disjoint blocks) of maximum size . We want to derive a bound on . (I conjecture that in fact r is bounded, e.g., r ≦ 4 – 4 is attained for the Fano plane, but all that has been proved so far (cf. [1], [2]) are bounds r < C.v for some C. Here we shall prove r < 5v2/3.)

Define a sequence of positive real numbers by q0 = Q · r2/v, , where l is determined by ql ≧ 6, , i.e.,

(The constant Q will be chosen later.) Define inductively sets Ai, Ki and collections as follows. Let

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1202 - 1204
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Lindner, C. C. and Phelps, K. T., A note on partial parallel classes in Steiner systems, Discr. Math. 24 (1978), 109–112.Google Scholar
2. Wang, S. P., On self orthogonal Latin squares and partial transversals of Latin squares, Ph.D. thesis, Ohio State University, Columbus, Ohio (1978).Google Scholar