The Voronov-Neopryatnaya-Dergachev graphs are two tetahedron-free (but not triangle-free graphs) on 372 and 972 vertices which have unit-distance embeddings with all vertices on a sphere and chromatic number 5. The smaller of these is embedded on the circumsphere of a regular icosahedron with a unit edge length, while the larger is embedded on the circumsphere of a great icosahedron with unit egde lengths.
These graphs are implemented in the Wolfram Language as GraphData["VoronovNeopryatnayaDergachevGraph372"] and GraphData["VoronovNeopryatnayaDergachevGraph972"], respectively.
de Grey (2026) subsequenctly constructed a 61-vertex graph with chromatic number 5 that has a unit-distance embedding
in 
and is tetrahedron-free.
With 61 Vertices." Geombinatorics 35,
2026.Voronov, V. A.; Neopryatnaya, A. M.; and Dergachev, E. A.
"Constructing 5-Chromatic Unit Distance Graphs Embedded in the Euclidean Plane
and Two-Dimensional Spheres." Disc. Math. 345, 113106 1-14, 2022.