Tree-Degenerate Graphs and Nested Dependent Random Choice

The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets $S$ in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of F\"uredi and of Alon, Krivelevich, and Sudakov showing that the maximum number of edges in an $n$-vertex graph not containing a fixed bipartite graph with maximum degree at most $r$ on one side is $O(n^{2-1/r})$. This was recently extended by Grzesik, Janzer and Nagy to the family of so-called $(r,t)$-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov, confirming a special case of a conjecture of Erd\H{o}s and Simonovits and of Sidorenko, showing that if $H$ is a bipartite graph that contains a vertex complete to the other part and $G$ is a graph then the probability that the uniform random mapping from $V(H)$ to $V(G)$ is a homomorphismis at least $\left[\frac{2|E(G)|}{|V(G)|^2}\right]^{|E(H)|}$. In this note, we introduce a nested variant of the dependent random choice lemma, which might be of independent interest. We then apply it to obtain a common extension of the theorem of Conlon, Fox, and Sudakov and the theorem of Grzesik, Janzer, and Nagy, regarding Tur\'an and Sidorenko properties of so-called tree-degenerate graphs.