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 Erdős 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 Turán and Sidorenko properties of so-called tree-degenerate graphs.