D-homogeneous and d-ultrahomogeneous linear spaces

Abstract

A linear space S is d-homogeneous if, whenever the linear structures induced on two subsets S1 and S2 of cardinality at most d are isomorphic, there is at least one automorphism of S mapping S1 onto S2. S is called d-ultrahomogeneous if each isomorphism between the linear structures induced on two subsets of cardinality at most d can be extended into an automorphism of S. We have proved in [11;] (without any finiteness assumption) that every 6-homogeneous linear space is homogeneous (that is d-homogeneous for every positive integer d). Here we classify completely the finite nontrivial linear spaces that are d-homogeneous for d ≥ 4 or d-ultrahomogeneous for d ≥ 3. We also prove an existence theorem for infinite nontrivial 4-ultrahomogeneous linear spaces. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 321–329, 2000