# Covering numbers - looking for a more combinatorial proof

For cardinals $$\mu$$, $$\kappa$$, $$\theta$$, and $$\sigma$$, the covering number $$cov(\mu,\kappa,\theta,\sigma)$$ is defined to be the minimum cardinality of a set $$P\subseteq [\mu]^{<\kappa}$$ such that every $$A\in [\mu]^{<\theta}$$ is covered by a union of fewer than $$\sigma$$ elements of $$P$$.

These numbers figure prominently in applications of pcf-theory.

It is a result of Gitik and Shelah (Theorem 1.5 of [GiSh:412], indexed in Shelah's archive) that for fixed cardinals $$\mu>\kappa$$, the set

$$\{cov(\mu,\kappa,\theta,\omega_1):\theta\leq\kappa\}$$

is finite.

Their proof is by contradiction, and involves transforming this into a question of pcf theory by using something called the cov vs. pp Theorem, and then using properties of pseudopowers to produce the contradiction.

(1) The translation'' into pseudopowers is the reason that the fourth component in the cov statement is uncountable, as that is a requirement of the cov vs. pp Theorem. It is open if that restriction is necessary.
(2) Gitik and Shelah point out that their result can be considered as a ZFC version of a result of Hajnal, which states that $$\{\mu^\theta:2^\theta<\mu\}$$ is finite.
Question: Is there a different'' proof of the result of Gitik and Shelah that works directly with the structures $$[\mu]^{<\kappa}$$, without needing to invoke the cov vs. pp Theorem to translate the question into the language of pseudopowers?
The hope would be that a proof through a different route would shed light on the role of $$\omega_1$$ in the Gitik-Shelah result. Is it there as an artifact of their proof, or is it really needed?