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

✍ Scribed by David A. Pike; Asiyeh Sanaei

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 129 KB
• Volume: 19
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20271

✦ Synopsis

In this article we study the n-existential closure property of the block intersection graphs of infinite t-(v, k, k) designs for which the block size k and the index k are both finite. We show that such block intersection graphs are 2-e.c. when 2 ≤ t ≤ k-1. When k = 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 k ≥ 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.