Existence of partial transversals

A Simple Condition for the Existence of Transversals

Hall’s Theorem is a basic result in Combinatorics which states that the obvious necesssary condition for a finite family of sets to have a transversal is also sufficient. We present a sufficient (but not necessary) condition on the sizes of the sets in the family and the sizes of their intersections so that a transversal exists. Using this, we prove that in a bipartite graph G (bipartition {A,B...

A Criterion for the Existence of Transversals of Set Systems

Nash-Williams [6] formulated a condition that is necessary and sufficient for a countable family A = (Ai)i∈I of sets to have a transversal. In [7] he proved that his criterion applies also when we allow the set I to be arbitrary and require only that ⋂ i∈J Ai = ∅ for any uncountable J ⊆ I. In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the ...

Another Form of a Criterion for the Existence of Transversals

In [2] we proved a necessary and sufficient condition for a family of sets to possess a transversal. We now prove a slightly more concrete version of this result, using the function q of [4].

A General Criterion for the Existence of Transversals

We present a necessary and sufficient condition for a family of sets to possess a transversal. Its form follows that of P. Hall's theorem: a family has a transversal if and only if it does not contain one of a set of 'forbidden' substructures.

Disjoint Common Partial Transversals of Two Families of Sets