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Claim. K. B. Chilakamarri, A 4-chromatic unit-distance graph with no triangles, Geombinatorics 4 (1995), no. 3, 64–76, constructs an infinite family of unit distance graphs in the plane with girth $4$ and chromatic number $4$, the smallest on $47$ vertices, as the site's commentary on Problem 705 reports. If its point sets have no unit distance besides the graphs' edges, no $k\le4$ makes every finite unit distance graph of girth at least $k$ $3$-colorable.

Covers. No $k\le4$ works. Nothing is settled for $k\ge5$; Wormald's graph of girth $5$ settles $k=5$ (Wormald 1979), and O'Donnell's dissertation every $k$ (O'Donnell 1999).

Depends on. No page of this wiki.

Standing. Claimed. The paper appeared in Geombinatorics, whose archive lists it in the issue of January 1995 and holds no copy online. The problem asks about the faithful graph of a point set, with an edge exactly when two points are at distance $1$, and whether this paper's point sets carry no unit distance besides the graphs' edges is not established from the paper, so refereed is not listed. The curator's label credits O'Donnell's dissertation, not this paper.

Dating. The journal's archive dates the issue January 1995; the day is a placeholder.