A Hall-type theorem for triplet set systems based on medians in trees

Given a collection C of subsets of a finite set X, let S C = ∪S∈CS. Philip Hall’s celebrated theorem [2] concerning ‘systems of distinct representatives’ tells us that for any collection C of subsets of X there exists an injective (i.e. one-to-one) function f : C → X with f(S) ∈ S for all S ∈ C if and and only if C satisfies the property that for all non-empty subsets C ′ of C we have | S C | ≥ |C |. Here we show that if the condition | S C | ≥ |C | is replaced by the stronger condition | S C | ≥ |C |+2, then we obtain a characterization of this condition for a collection of 3-element subsets of X in terms of the existence of an injective function from C to the vertices of a tree whose vertex set includes X and that satisfies a certain median condition. We then describe an extension of this result to collections of arbitrary-cardinality subsets of X. 1. First result Given a tree T = (V, E) and a subset S of V of size 3, say S = {x, y, z}, consider the path in T connecting x, y, the path connecting x, z and the path connecting y, z. There is a unique vertex that is shared by these three paths, the median vertex of S in T , denoted medT (S). Theorem 1.1. Let X be a finite set, and suppose that C ⊆ ( X 3 ) , and ⋃ C = X. The following are equivalent: (1) There exists a tree T = (V, E) with X ⊆ V for which the function S 7→ medT (S) from C to V is injective. (2) There exists a tree T = (V, E) with X as its set of leaves, and all its other vertices of degree 3, for which the function S 7→ medT (S) from C to the set of interior vertices of T is injective. (3) C satisfies the following property. For all non-empty subsets C ′ of C we have: (1) | ⋃ C | ≥ |C |+ 2. In order to establish Theorem 1.1 we first require a lemma. 1991 Mathematics Subject Classification. 05C05.