We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the author from [13] to show that these equivalence relations are invariantly universal, in the sense of [3], and thus in particular complete analytic. We also introduce a new kind of Borel reducibility relation for standard Borel G-spaces, which requires the preservation of stabilizers, and explain its connection with the notion of full embeddings commonly considered in category theory.
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In [5] H. A. Wilbrink proved that a certain class of Minkowski planes induce nearaffine planes. Of course, this class contains all Minkowski planes over fields. But only Minkowski planes of order 3 and 4 induce nearaffine planes which are also Minkowski planes (moreover they are affine planes, too.
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