Homogeneous and ultrahomogeneous linear spaces (Q1273672)

A linear space \(S\) is called \(d\)-homogeneous, if whenever the linear structures induced on two subsets \(S_1\) and \(S_2\) of cardinality at most \(d\) are isomorphic, there is at least one automorphism \(S\) mapping \(S_1\) onto \(S_2\); if every isomorphism from \(S_1\) to \(S_2\) can be extended to an automorphism of \(S\), we shall say that \(S\) is \(d\)-ultrahomogeneous. \(S\) is called homogeneous (resp. ultrahomogeneous) if it is \(d\)-homogeneous (resp. ultrahomogeneous) for all positive integers \(d\). In this paper it is obtained a complete classification of all homogeneous (resp. ultrahomogeneous) linear spaces \(S\), without making any finiteness assumption on the number of points of \(S\). Theorem. Any 6-homogeneous linear space is homogeneous. Any homogeneous linear space is one of the following: (1) a linear space reduced to a single point (and no line) or to a single line; (2) a linear spaces all of whose lines have exactly two points; (3) the Desarguesian affine plane \(AG(2,3)\); (3) one of the Desarguesian projective planes \(PG(2,2)\), \(PG(2,3)\) or \(PG(2,4)\). Any homogeneous linear space is ultrahomogeneous, except \(AG(2,3)\) and \(PG(2,4)\). Note also that \(PG(2,5)\) is not homogeneous, but it is \(5\)-homogeneous.