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 have a common neighborhood almost as large as in the random graph of the same edge-density. There are two well-known applications of this lemma. The first is a theorem of Füredi [Combinatorica, 11 (1991), pp. 75–79] and Alon, Krivelevich, and Sudakov [Combin. Probab. Comput., 12 (2003), pp. 477–494] 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 [J. Combin. Theory Ser. B, 156 (2022), pp. 299–309] to the family of so-called \((r,t)\)-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov [Geom. Funct. Anal., 20 (2010), pp. 1354–1366], 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 that is completely joined 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 homomorphism is at least \(\left [\frac{2|E(G)|}{|V(G)|^2}\right ]^{|E(H)|}\). In this paper, 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.