Abstract
Among the classes of models which demonstrate the independence of Hilbert's axioms of spatial affine geometry there is one which is structurally interesting also in its own right. Its models represent the spaces in which not all triples of non-collinear points lie in a unique plane. Barring some entirely trivial variations 39 models exist. They all are finite, and each of their lines is incident with exactly two points. It turns out that 37 of the models are isomorphic to the 37 non-empty, triangle-free graphs with six vertices. The collection of these graphs exhibits a so far unexplained symmetry property.
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