Steiner triple systems and existentially closed graphs A.D. Forbes, M.J. Grannell and T.S. Griggs Department of Pure Mathematics The Open University Walton Hall Milton Keynes MK7 6AA UNITED KINGDOM tonyforbes@ltkz.demon.co.uk m.j.grannell@open.ac.uk t.s.griggs@open.ac.uk Submitted Dec 20, 2004; Accepted: Apr 7, 2005; Published: Aug 30, 2005 Mathematics Subject Classifications: 05C99, 05B07 Abstract We investigate the conditions under which a Steiner triple system can have a 2- or 3-existentially closed block intersection graph. 1 Introduction A graph G =(V,E), where V is the set of vertices and E is the set of edges, is said to be n-existentially closed, or n-e.c., if for every n-element subset S of V , and for every subset T of S , there exists a vertex x ∈ S which is adjacent to every vertex in T , and is not adjacent to any vertex in S \ T . These graphs were first studied by Caccetta, Erd˝ os and Vijayan, [6], although Erd˝ os and R´ enyi, [8], had previously proved the interesting result that for any fixed value of n, almost all graphs are n-e.c. But relatively few specific examples of n-e.c. graphs are known for n ≥ 2. A strongly regular graph SRG(v, k, λ, μ) is a regular graph of degree k on v vertices with the property that every pair of adjacent vertices has λ common neighbours and every pair of non-adjacent vertices have μ common neighbours. An important class of strongly regular graphs are the Paley graphs. The Paley graph of order q , where q ≡ 1 (mod 4) is a prime power, is the graph with vertex set GF(q ), the Galois field of order q , and the edge set is the set of pairs {x, y } where x − y is a square. It is an SRG(q, (q −1)/2, (q −5)/4), (q −1)/4). Ananchuen and Caccetta, [1], proved that all Paley graphs with at least 29 vertices are 3-e.c.; see also [3] and [5]. In [2], Baker, Bonato and Brown constructed 3-e.c. strongly regular graphs SRG(q 2 , (q 2 − 1)/2, (q 2 − 5)/4, (q 2 − 1)/4) the electronic journal of combinatorics 12 (2005), #R42 1