Nonincident points and blocks in designs

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