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 ⋃ C = ∪ S ∈ C S . Philip Hall’s celebrated theorem [P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935) 26–30] 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 | ⋃ C ′ | ≥ | C | . Here, we show that if the condition | ⋃ C ′ | ≥ | C ′ | is replaced by the stronger condition | ⋃ 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 which satisfies a certain median condition. We then describe an extension of this result to collections of arbitrary-cardinality subsets of X .