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

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 < 5v 2/3.) Define a sequence of positive real numbers by q 0 = Q · r 2/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