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

摘要

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.