Is there some $k$ such that if $G$ has girth $\geq k$ (i.e. $G$ contains no cycles of length $<k$) then $\chi(G)\leq 3$?

OPEN

Let $G$ be a finite unit distance graph in $\mathbb{R}^2$ (i.e. the vertices are a finite collection of points in $\mathbb{R}^2$ and there is an edge between two points f and only if the distance between them is $1$).

Is there some $k$ such that if $G$ has girth $\geq k$ (i.e. $G$ contains no cycles of length $<k$) then $\chi(G)\leq 3$?

The maximal value of $\chi(G)$ (without a girth condition) is the Hadwiger-Nelson problem. There are unit distance graphs (e.g. the Moser spindle) with $\chi(G)=4$ of girth $3$. de Grey [dG18] has constructed a unit distance graph $G$ with $\chi(G)=5$. (I do not know what the largest girth achieved is by these recent constructions.)

Wormald [Wo79] has constructed a unit distance graph with $\chi(G)=4$ and girth $5$, with $6448$ vertices. O'Donnell [OD94] has constructed a unit distance graph with $\chi(G)=4$ and girth $4$, with $56$ vertices. Chilakamarri [Ch95] has constructed an infinite family of unit distance graphs with $\chi(G)=4$ and girth $4$, the smallest of which has $47$ vertices.