Supersaturation for hereditary properties

Let $\mathcal{F}$ be a collection of $r$-uniform hypergraphs, and let $0 < p < 1$. It is known that there exists $c = c(p,\mathcal{F})$ such that the probability of a random $r$-graph in $G(n,p)$ not containing an induced subgraph from $\mathcal{F}$ is $2^{(-c+o(1)){n \choose r}}$. Let each graph in $\mathcal{F}$ have at least $t$ vertices. We show that in fact for every $\epsilon > 0$, there exists $\delta = \delta (\epsilon, p,\mathcal{F}) > 0$ such that the probability of a random $r$-graph in $G(n,p)$ containing less than $\delta n^t$ induced subgraphs each lying in $\mathcal{F}$ is at most $2^{(-c+\epsilon){n \choose r}}$. This statement is an analogue for hereditary properties of the supersaturation theorem of Erd\H{o}s and Simonovits. In our applications we answer a question of Bollob\'as and Nikiforov.Comment: 5 pages, submitted to European Journal of Combinatoric

Consistent random vertex-orderings of graphs

Given a hereditary graph property $\mathcal{P}$, consider distributions of random orderings of vertices of graphs $G\in\mathcal{P}$ that are preserved under isomorphisms and under taking induced subgraphs. We show that for many properties $\mathcal{P}$ the only such random orderings are uniform, and give some examples of non-uniform orderings when they exist