Existential closure of block intersection graphs of infinite designs having finite block size and index

In this article we study the n ‐existential closure property of the block intersection graphs of infinite t ‐( v, k , λ) designs for which the block size k and the index λ are both finite. We show that such block intersection graphs are 2‐ e.c . when 2⩽ t ⩽ k − 1. When λ = 1 and 2⩽ t ⩽ k , then a necessary and sufficient condition on n for the block intersection graph to be n ‐ e.c. is that n ⩽min{ t , ⌊( k − 1)/( t − 1)⌋ + 1}. If λ⩾2 then we show that the block intersection graph is not n ‐ e.c . for any n ⩾min{ t + 1, ⌈ k / t ⌉ + 1}, and that for 3⩽ n ⩽min{ t , ⌈ k / t ⌉} the block intersection graph is potentially but not necessarily n ‐ e.c . The cases t = 1 and t = k are also discussed. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 85–94, 2011