A family [Fscr ] of functions from ℝn
to ℝ is k-point separating if, for every k-subset
S
of ℝn, there is a
function f∈[Fscr ] such that f is one-to-one on S.
The paper shows that, if the functions are required to be linear
(or smooth), then a minimum k-point separating family [Fscr ] has
cardinality
n(k−1). In the linear case, this
result is extended to a larger class of fields including all infinite fields
as well as some finite fields (depending
on k and n). Also, some partial results are obtained
for
continuous functions on ℝn, including the case
when
k is infinite. The proof of the main result is based on graph
theoretic results that have some interest in their
own right. Say that a graph is an n-tree if it is a union of
n edge-disjoint spanning trees. It is shown that
every graph with k[ges ]2 vertices and n(k−1)
edges has a non-trivial subgraph which is an n-tree. A
determinantal criterion is also established for a graph with k
vertices
and n(k−1) edges to be an n-tree.