Discrete Mathematics 129 (1994) 29-41 North-Holland 29 Infinite linear spaces Peter J. Cameron School of Mathematical Sciences, QMW, University of London, Mile End Road, London El 4NS. UK Received 12 June 1991 Revised 27 October 1991 1. Introduction There is no theory of infinite linear spaces comparable to the enormous amount known about finite linear spaces. This is due to two contrasting factors. First, techniques which are crucial in the finite case (notably counting) are not available. Second, infinite linear spaces are too easy to construct; instead of having to force our configurations to ‘close up’, we just continue adding points and lines infinitely often! The result is a proliferation of examples without any set of tools to deal with them. So there will be several unanswered questions here! I have concentrated in this survey on infinite analogues of finite results and examples, but I also mention things which have no finite analogue. Section 2 describes a couple of simple free constructions which nevertheless provide us with evidence for the proliferation of examples mentioned. Even projective planes are wild, as we see in Section 3. In Section 4, classical characterisations of some well-known structures are given: this is one area where the infinite is not so different from the finite. The next two sections describe some results about automorphisms: in Section 5 we see Simon Thomas’ dichotomy between doubly transitive and triangle- transitive Steiner triple systems, and in Section 6 we consider generalizing the orbit theorem (Block’s lemma) to the infinite. Sections 7 and 8 consider a couple of miscellanea: spreads, and generalized quadrangles. The terminology usually follows Dembowski [6]. 2. Finite line size Infinite linear spaces with any given infinite number of points, and any given finite number of points per line, exist in great profusion. Indeed, such objects can be Correspondence to: Peter J. Cameron, School of Mathematical Sciences, Queen Mary and Westfield College, University of London, Mile End Road, London El 4NS, UK. 0012-365X/94/$07.00 0 1994-Elsevier Science B.V. All rights reserved SSDI 0012-365X(92)00503-B brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector