  We construct a new family of homomorphisms from Specht modules into Foulkes
modules for the symmetric group. These homomorphisms are used to give a
combinatorial description of the minimal partitions (in the dominance order)
which label irreducible characters appearing as summands of the characters of
Foulkes modules. The homomorphisms are defined using certain families of
subsets of the natural numbers. These families are of independent interest; we
prove a number of combinatorial results concerning them.
