Nonincident points and blocks in designs

In this paper, we study the problem of finding the largest integer ss for which there exists a set of ss points and ss blocks in a balanced incomplete block design such that none of the ss points lie on any of the ss blocks. We investigate this problem for two types of BIBDs: projective planes and Steiner triple systems. For a Steiner triple system on vv points, we prove that s≤(2v+5−24v+25)/2, and we determine necessary and sufficient conditions for equality to be attained in this bound. For a projective plane of order qq, we prove that s≤1+(q+1)(q−1), and we show that equality can be attained in this bound whenever qq is an even power of two.