Borel Homomorphisms from Forests to Kneser Graphs

We answer a recent question of Cs\'oka and Vidny\'anszky [arXiv:2407.10006] and give an alternate proof of one of their results. The subject of both is which finite graphs admit factor of i.i.d. homomorphisms from the 3-regular tree. We then give yet another proof of the result in the Borel setting which leads to the following: For each $d > 2$ and $k \in \mathbb{N}$, there is a Borel hyperfinite $d$-regular forest $G$ and a finite graph with chromatic number $k$, $H$, so that $G$ does not admit a Borel homomorphism to $H$. All of this is tied together by a focus on the case when the target graph $H$ is a (subgraph of a) Kneser graph.