Embedding Induced Bounded Degree Graphs

We prove a sparse embedding theorem for induced embeddings of bounded-degree graphs. The theorem applies to pairs $G\subseteq \Gamma$: the graph $G$ supplies the positive edges of the target graph, while the ambient graph $\Gamma$ supplies the induced constraints which must be avoided. Its main feature is that these two types of constraints are kept separate throughout the embedding process. As an application, we show that, for every fixed $\Delta,r\ge2$, there are constants $A,C>0$ such that every $n$-vertex graph $H$ with maximum degree at most $\Delta$ satisfies $r_{\text{ind}}(H;r)\le C n^{\Delta+2}(\log n)^A$. This improves the exponent in the polynomial bound of Conlon, Fox and Zhao for bounded-degree induced Ramsey numbers. The proof combines the aforementioned embedding theorem with a sparse random transference argument, in which the random host is used only to certify robust deterministic hypotheses for every colour class.