On Existentially Complete Triangle-Free Graphs

For a positive integer k, we say that a graph is k-existentially complete if for every 0 <= a <= k, and every tuple of distinct vertices x(1), horizontal ellipsis , x(a), y(1), horizontal ellipsis , y(k-a), there exists a vertex z that is joined to all of the vertices x(1), horizontal ellipsis , x(a) and to none of the vertices y(1), horizontal ellipsis , y(k-a). While it is easy to show that the binomial random graph G(n,1/2) satisfies this property (with high probability) for k = (1 - o(1)) log(2)n, little is known about the "triangle-free" version of this problem: does there exist a finite triangle-free graph G with a similar "extension property"? This question was first raised by Cherlin in 1993 and remains open even in the case k = 4. We show that there are no k-existentially complete triangle-free graphs on n vertices with k>8, for n sufficiently large.