Symmetric group character degrees and hook numbers

In this article we prove the following result: for any two natural numbers k and ℓ, and for all sufficiently large symmetric groups Sn, there are k disjoint sets of ℓ irreducible characters of Sn, such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moretó in [5]. The methods employed here are based upon the duality between irreducible characters of the symmetric groups and the partitions to which they correspond. Consequently, the paper is combinatorial in nature.