Borel Kernels in Borel Directed Graphs

We prove that there is a Borel quasi-kernel in any locally countable Borel directed graph with finite Borel chromatic number. We prove that the Borel chromatic number of a Borel directed graph with bounded out-degree $n$ is either infinite or less than or equal to $\frac{(n+1)(n+2)}{2}$. This is an alternative proof of Palamourdas’ theorem.