Towards Erdős-Hajnal property for dp-minimal graphs

We introduce the notion of strongly $\left(\genfrac{}{}{0px}{}{k}{2}\right)$-free graphs, which contain dp-minimal graphs. We show that under some sparsity assumption, given a rainbow $\left(\genfrac{}{}{0px}{}{k}{2}\right)$-free blockade we can find a rainbow $\left(\genfrac{}{}{0px}{}{k-1}{2}\right)$-free blockade. This might serve as an intermediate step towards Erd\H os-Hajnal property for dp-minimal graphs.