Ultrahomogeneous Semilinear Spaces

A semilinear space S is a non-empty set of elements called points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points. Two points p and q are said to be collinear if p 6ˆ q and if the pair f p; qg is contained in a line; the line containing two collinear points p and q will be denoted by pq. The degree of a point p is the cardinal number of the set of lines through p, the neighbourhood of p is the set of all points of S which are collinear with p, and a neighbour of p is an element of its neighbourhood. The collinearity graph of a semilinear space S is the graph whose vertices are the points of S and in which two vertices are adjacent if and only if the corresponding points in S are collinear; S is said to be connected if its collinearity graph is connected. The connected components of S are the connected components of its collinearity graph. Semilinear spaces are a common generalization of graphs (when all lines have exactly two points) and of linear spaces (when any pair of points is contained in exactly one line). A semilinear space which is neither a graph nor a linear space will be called proper. If S 0 is a non-empty subset of S, the semilinear structure induced on S 0 is the semilinear space whose points are those of S 0 and whose lines are the intersections of S 0 with all the lines of S having at least two points in S 0. Given a positive integer d , a semilinear space S is said to be d-homogeneous if, whenever the semilinear structures induced on two subsets S1 and S2 of S of cardinality at most d are isomorphic, there is at least one automorphism of S mapping S1 onto S2; if every isomorphism from S1 to S2 can be extended to an automorphism of S, we shall say that S is d-ultrahomogeneous. A semilinear space S is called homogeneous if S is d-homogeneous for every positive integer d , and is ultrahomogeneous if S is d-ultrahomogeneous for every d. The notions of homogeneity and ultrahomogeneity can be de®ned more generally in any class of relational structures (see Cameron [3]). Historically, these notions arise from the Helmholtz±Lie principle of `free mobility of rigid bodies' (1880), a property common to the motion groups of Euclidean and nonEuclidean spaces: in such spaces, every (®nite) set of points can be carried by motion onto any congruent one (see Freudenthal [10] for more details). Georg Cantor [4] pointed out in 1895 that the ordered set of rational numbers …Q; <† is ultrahomogeneous.