In this paper, we study the problem of finding the largest possible set of s points and s blocks in a balanced incomplete block design, such that 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 ≤ (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 q, we prove that s ≤ 1 + (q + 1) ( √ q − 1) and we also show that equality can be attained in this bound whenever q is an even power of two.
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