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Claim. P. O'Donnell, A triangle-free 4-chromatic graph in the plane, Geombinatorics 4 (1994), no. 1, 23–29, constructs a unit distance graph in the plane on $56$ vertices with girth $4$ and chromatic number $4$, as the site's commentary on Problem 705 reports. If its point set has no unit distance besides the graph's 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 July 1994 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 $1$, while O'Donnell's own definition of a unit distance graph in his 1999 dissertation allows further unit distances between non-adjacent vertices. Whether this paper's point set has none is not established from the paper, so refereed is not listed. The curator's label credits the dissertation, not this paper.

Dating. The journal's archive dates the issue July 1994; the day is a placeholder.