On existentially complete triangle-free graphs

Abstract

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 x₁, …, x_a, y₁, …, y_k−a, there exists a vertex z that is joined to all of the vertices x₁, …, x_a and to none of the vertices y₁, …, 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)) log₂n, 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 $k\, > \,{{8\log \,n} \over {\log \,\log n}}$, for n sufficiently large.