INTRODUCTION

A yery famous theorem (associated with the names Hilbert, von Staudt, Veblen and Young) characterizes projective spaces of dimension greater than 2 as linear incidence systems satisfying a certain (variously named) axiom. By the term “characterization”, one means a complete classification in terms of division rings. This famous characterization theorem fully displays the spirit of synthetic geometry in that one obtains an exact and elaborate structure with many subspaces from a few simple axioms mentioning only points and lines.

More than three decades later F. Buekenhout and the author obtained a characterization of polar spaces of rank more than 2 in terms of a similar set of very simple axioms concerning only points and lines. But this time the characterization rested on a considerably more involved theory of Veldkamp and Tits, where, in effect, the really difficult work was done. Indeed Tits' work on polar spaces (as axiomatized by him) was a part of his monumental classification of buildings of spherical type of rank greater than 2. The Buekenhout-Shult polar space theorem could then be seen as a characterization of the buildings of types C and D in terms of axioms involving only two types of varieties of the building. The question was then raised (see [20]) whether similar axiomatically simple “point-line” characterizations could be obtained for all the buildings of spherical type of rank at least 3.