Borel Homomorphisms from Forests to Kneser Graphs

TL;DR

This work investigates Borel and measurable homomorphisms from 3-regular acyclic Borel forests to finite graphs, focusing on Kneser-type targets and fiid phenomena. It provides an alternate approach to Csóka–Vidnyánszky results and shows that for each $d>2$ and $k∈ olinebreak\mathbb{N}$ there exists a Borel hyperfinite $d$-regular forest $G$ and a finite graph $H$ with χ(H)=k such that no Borel homomorphism $G→H$ exists; the hyperfinite version is achieved via a game-based, Baire-measurable construction using Hom$^e(T_d,\ olinebreak\mathcal{H})$ and a comeager subgraph $\mathcal{H}$. The paper also develops preliminaries on fiid, Bernoulli shifts, and fractional/chromatic properties of Kneser and Schrijver graphs, enabling alternate proofs and new insights into odd girth constraints. Overall, the results connect descriptive set theory, fiid combinatorics, and classical graph coloring to reveal new obstructions to Borel homomorphisms and to suggest further questions about girth, hyperfiniteness, and small-target graphs.

Abstract

We answer a recent question of Csóka and Vidnyánszky [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.

Figures (2)

• Figure 1: The 2-ball around the identity in $\text{Cay}(\mathbb{Z}_2^{*3},S_3)$ with our edge labels. In the games $G(x,a_2a_0)$, Alice is responsible for labeling the red node (and all the other nodes in that sextant), Bob for the blue nodes (and all the other nodes in those five sextants), and the white nodes other than the identity are given the label 0.
• Figure 2: Playing two copies of Bob against eachother. As in Figure \ref{['fig:game']}, $d = 3$, $l = 1$, and $\tau = a_2a_0$. The two Bobs agree on edge labels, but vertices are labeled according to the unprimed Bob's perspective. The unprimed Bob is responsible for labeling the blue vertices. From his perspective, everything to the right of the dotted line is being labeled by Alice. In reality we label $\tau$ with $m$ and the remaining nonidentity white nodes 0, while the primed Bob labels the green vertices. The fact that each $f_{i,j}$ is an involution can be used to see that the primed Bob agrees with the unprimed Bob, modulo shifting by $\sigma$, about which element of $V(G)$ is being constructed.

Theorems & Definitions (45)

• Definition 1.1
• Theorem 1.2: csoka_vidnyanszky
• Corollary 1.3: csoka_vidnyanszky
• proof
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Corollary 1.7
• Definition 2.1
• Definition 2.2