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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.

Comments:

(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?

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