The purpose of this paper is to demonstrate that a number of properties of independence spaces are of finite character, thus making it possible to easily generalise known theorems for finite spaces, or matroids, to independence spaces on infinite sets.

Trees are basic in graph theory and its applications to many fields, such as chemistry, electric network theory, and the theory of games. König [7; 47-48] gives an interesting historical account of independent discoveries of trees by Kirchhoff, Cayley, Sylvester, Jordan, and others who were working in a variety of fields.

There are many equivalent ways of defining trees, the most common being: A tree is a graph which is connected and has no circuits. Figure 1 shows the trees with up to six vertices; those with up to 10 vertices can be found in Harary [5; 233–234]. Some other possible definitions of a tree are the following: (1) A tree is a graph which is connected and has one more vertex than edge. (2) A tree is a graph which has no circuits and has one more vertex than edge. For these and other characterizations see Berge [4; 152ff.] and Harary [5; 32ff.]. A less common definition is by induction: The graph consisting of a single vertex is a tree, and a tree with n + 1 vertices is obtained from a tree with n vertices by adding a new vertex adjacent to exactly one other vertex.

Plummer (see [2; p. 69]) conjectured that the square of every block is hamiltonian, and this has just been proved by Fleischner [1]. It was shown by Karaganis [3] that the cube of every connected graph, and hence the cube of every tree, is hamiltonian. Our present object is to characterize those trees whose square is hamiltonian in three equivalent ways.

We follow the terminology and notation of the book [2]. In particular, the following concepts are used in stating our main result. A graph is hamiltonian if it has a cycle containing all its points. The graph with the same points as G, in which two points are adjacent if their distance in G is at most 2, is denoted by G2 and is called the square of G. The subdivision graph S(G) is formed (Figure 1) by inserting a point of degree two on each line of G.

In [1], A. H. Stone proved that for a cardinal number k ≥ 1 a set with a transitive relation can be partitioned into k cofinal subsets provided each element of the set has at least k successors. Using methods quite different from those of Stone, we show that for k ≥ ℵ0 the same condition on successors guarantees that a set on which there are defined not more than k transitive relations can be partitioned into k sets each of which is cofinal with respect to each of the relations. We also show that such a partition exists even if some of the relations are not transitive as long as the non-transitive relations have no more than k elements.

A tree is a connected graph that has no cycles. If x is any endnode of a tree, then the limb determined by x is the unique path that joins x with the nearest node other than x that does not have degree two in the tree; let l(x) denote the length of this path. (For definitions and results not given here see [2] or [3].) Different endnodes determine different limbs with one exception; when the tree is a path then both endnodes determine the same limb, namely, the tree itself. Our object here is to investigate the distribution of the length of limbs of trees Tn chosen at random from the set of nn-2 trees with n labelled nodes; in particular, it will follow from our results that the length of the longest limb in most trees Tn is approximately log n when n is large.