Borel Homomorphisms from Forests to Kneser Graphs

TL;DR

This paper investigates Borel homomorphisms from forests to Kneser graphs, providing new proofs and demonstrating limitations of such homomorphisms in the Borel setting, especially for hyperfinite forests and graphs with specific chromatic numbers.

Contribution

It offers alternative proofs of existing results and establishes new limitations on Borel homomorphisms from forests to certain graphs, including Kneser graphs, in the Borel setting.

Findings

Existence of Borel hyperfinite forests with no Borel homomorphism to certain graphs.

New proofs of results related to factor of i.i.d. homomorphisms from the 3-regular tree.

Limitations on Borel homomorphisms to graphs with prescribed chromatic number.

Abstract

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.