Title:Universality of graphs with few triangles and anti-triangles

Abstract: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 ${\mathcal G}_{n,1/2}$ is, in particular, 3-random-like, this can be viewed as a weak version of quasirandomness. 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. We then show that for larger subgraphs, 3-random-like sequences demonstrate a completely different behaviour. We prove that for every graph $H$ on $n\geq R(10,10)$ vertices there exist 3-random-like graphs without an induced copy of $H$. Moreover, we prove that for every $\ell$ there are 3-random-like graphs which are $\ell$-universal but not $m$-universal when $m$ is sufficiently large compared to $\ell$.