Universality of Graphs with Few Triangles and Anti-Triangles

We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph G(n,1/2) is, in particular, 3-random-like, this can be viewed as a weak version of quasi-randomness. We first show that 3-random-like graphs are 4-universal, that is, they contain induced copies of all 4-vertex graphs. This settles a question of Linial and Morgenstern [10]. We then show that for larger subgraphs, 3-random-like sequences demonstrate completely different behaviour. We prove that for every graph H on n >= 13 vertices there exist 3-random-like graphs without an induced copy of H. Moreover, we prove that for every l there are 3-random-like graphs which are l-universal but not m-universal when m is sufficiently large compared to l.