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

Abstract

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