a family F as above of minimal size satisfies $|F|=cov(\aleph_\omega,\aleph_\omega,\aleph_\omega,2)=pp(\aleph_\omega)=cof([\aleph_\omega]^\omega)\ge \aleph_{\omega+1}$.
Edit: following Ali's suggestion, here are more details. $$cov(\lambda,\theta,\kappa,\sigma):=\min\left{ cov(\lambda,\theta,\kappa,\sigma):=min{ |\mathcal F| : \mathcal F\subseteq [\lambda]^{<\theta} s.t. \forall A\in [\lambda]^{<\kappa}\exists\mathcal A\in[\mathcal F]^{<\sigma}(A\subseteq\bigcup\mathcal A)\right}$$ A)}$$The definition is due to Shelah, of course. Note that cov(\lambda,\kappa,\kappa,2)=cf([\lambda]^{<\kappa},\subseteq). The definition of pp, may be found in several places; for a crush treatment, see for example: http://papers.assafrinot.com/?num=5 . Let \lambda denote a singular cardinal. It is always the case that \lambda^+\le pp(\lambda)\le cov(\lambda^+,\lambda,cf(\lambda)^+,2)\le cf([\lambda]^{cf(\lambda)},\subseteq). Now, consider the preceding three cardinal invaritans (of \lambda). Shelah proved that if \lambda is the least (singular) cardinal for which any of the three is greater than \lambda^+, then all three are equal. In particular, the openning equation that I gave (concerning \aleph_\omega) holds. See [Sh:E12] for pointers to Shelah's works on pp VS. cov''. Edit2: I see that the definition of cov is rendered incorrectly. The definition may be found here as well: http://papers.assafrinot.com/?num=1 . See (the proof of) Lemma 3.4 from there, and the subsequent works on this subject: 1. http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jsl/1164060456 2. http://journals.impan.pl/cgi-bin/doi?fm205-1-3 4 added 15 characters in body a family F as above of minimal size satisfies |F|=cov(\aleph_\omega,\aleph_\omega,\aleph_\omega,2)=pp(\aleph_\omega)=cof([\aleph_\omega]^\omega)\ge \aleph_{\omega+1}. Edit: following Ali's suggestion, here are more details.$$cov(\lambda,\theta,\kappa,\sigma):=\min\left{ |\mathcal F| : \mathcal F\subseteq [\lambda]^{<\theta} s.t. \forall A\in [\lambda]^{<\kappa}\exists\mathcal A\in[\mathcal F]^{<\sigma}(A\subseteq\bigcup\mathcal A)\right}$$The definition is due to Shelah, of course. Note that cov(\lambda,\kappa,\kappa,2)=cf([\lambda]^{<\kappa},\subseteq). The definition of pp, may be found in several places; for a crush treatment, see for example: http://papers.assafrinot.com/?num=5 . Let \lambda denote a singular cardinal. It is always the case that \lambda^+\le pp(\lambda)\le cov(\lambda^+,\lambda,cf(\lambda)^+,2)\le cf([\lambda]^{cf(\lambda)},\subseteq). Now, consider the preceding three cardinal invaritans (of \lambda). Shelah proved that if \lambda is the least (singular) cardinal for which any of the three is greater than \lambda^+, then all three are equal. In particular, the openning equation that I gave (concerning \aleph_\omega) holds. See [Sh:E12] for pointers to Shelah's works on pp VS. cov''. Edit2: I see that the definition of cov is rendered incorrectly. The definition may be found here as well: http://papers.assafrinot.com/?num=1 . See (the proof of) Lemma 3.4 from there, and the subsequent works on this subject: 1. http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jsl/1164060456 2. http://journals.impan.pl/cgi-bin/doi?fm205-1-3 3 added 388 characters in body; added 3 characters in body; added 5 characters in body a family F as above of minimal size satisfies |F|=cov(\aleph_\omega,\aleph_\omega,\aleph_\omega,2)=pp(\aleph_\omega)=cof([\aleph_\omega]^\omega)\ge \aleph_{\omega+1}. Edit: following Ali's suggestion, here are more details.$$cov(\lambda,\theta,\kappa,\sigma):=\min\left{ |\mathcal F| : \mathcal F\subseteq [\lambda]^{<\theta} s.t. \forall A\in [\lambda]^{<\kappa}\exists\mathcal A\in[\mathcal F]^{<\sigma}(A\subseteq\bigcup\mathcal A)\right} The definition is due to Shelah, of course. Note that $cov(\lambda,\kappa,\kappa,2)=cf([\lambda]^{<\kappa},\subseteq)$. The definition of $pp$, may be found in several places; for a crush treatment, see for example: http://papers.assafrinot.com/?num=5 . Let $\lambda$ denote a singular cardinal. It is always the case that $\lambda^+\le cov(\lambda,\lambda,cf(\lambda),2)\le cov(\lambda^+,\lambda,cf(\lambda)^+,2)\le cf([\lambda]^{cf(\lambda)},\subseteq)$. Now, consider the preceding three cardinal invaritans (of $\lambda$). Shelah proved that if $\lambda$ is the least (singular) cardinal for which any of the three is greater than $\lambda^+$, then all three are equal. In particular, the openning equation that I gave (concerning $\aleph_\omega$) holds. See [Sh:E12] for pointers to Shelah's works on pp VS. cov''.
Edit2: I see that the definition of $cov$ is rendered incorrectly. The definition may be found here as well: http://papers.assafrinot.com/?num=1 . See (the proof of) Lemma 3.4 from there, and the subsequent works on this subject: 1. http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jsl/1164060456 2. http://journals.impan.pl/cgi-bin/doi?fm205-1-3