Homogeneity of abstract linear spaces

TL;DR

This paper investigates the homogeneity and universality properties of abstract linear spaces, including projective planes, and establishes conditions under which certain such spaces are homogeneous or universal.

Contribution

It proves the homogeneity of the smallest projective planes and, assuming the continuum hypothesis, constructs a universal homogeneous projective plane of size 91, and relates the existence of a generic countable linear space to a longstanding conjecture.

Findings

Smallest projective planes are homogeneous.

Under continuum hypothesis, a universal homogeneous projective plane exists.

Existence of a generic countable linear space is linked to a conjecture about finite linear spaces.

Abstract

We discuss homogeneity and universality issues in the theory of abstract linear spaces, namely, structures with points and lines satisfying natural axioms, as in Euclidean or projective geometry. We show that the two smallest projective planes (including the Fano plane) are homogeneous and, assuming the continuum hypothesis, there exists a universal projective plane of cardinality $\aleph_1$ that is homogeneous with respect to its countable and finite projective subplanes. We also show that the existence of a generic countable linear space is equivalent to an old conjecture asserting that every finite linear space embeds into a finite projective plane.