# $\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$

Let me first recall some pretty standard notations:

• $\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$;
• $\mathfrak{b}$ is the bounding number;
• $\mathfrak{s}$ is the splitting number.

It is well-known that there are no relations in ZFC between those numbers. For example, the classical results state that the following inequalities are relatively consistent:

• $\omega_1=\mathfrak{s}<\mathfrak{b}=\omega_2$ (Balcar-Pelant-Simon '80);
• $\omega_1=\mathfrak{b}<\mathfrak{s}=\omega_2$ (Shelah '84);
• $\omega_1=\text{cov}(\mathcal{M})<\mathfrak{b}=\omega_2$ (Bartoszyński '96);
• $\omega_1=\mathfrak{b}<\text{cov}(\mathcal{M})=\omega_2$ (Miller '81).

However, I cannot find any quite straightforward consistency results concerning relations between $\mathfrak{s}$ and $\text{cov}(\mathcal{M})$, so I would be very grateful if you could provide me any references for them.

And what I am in fact interested the most are results showing that for any reasonable (whatever that means) triple $(\kappa,\lambda,\mu)$ of uncountable cardinals and a cardinal $\nu$ greater than any from the triple, there is a model of ZFC for which $\mathfrak{s}=\kappa,\mathfrak{b}=\lambda,\text{cov}(\mathcal{M})=\mu,\nu=2^\omega$. I am aware that such general models may still be unknown (provided they do exist at all), so partial results are also welcome.

I need those results to be referred to in a paper, so I am mainly interested in references where one can find proofs.

Thank you very much for any answer and please accept my apologies if the question is irrelevant for MO or my post contains mistakes.

• I don't think there is any provable relation between $\mathfrak{s}$ and $\mathrm{cov}(\mathcal{M})$. The reason is that in an extension adding many Cohen reals, the covering number is large and the splitting number is small, while the reverse happens in an extension adding many Mathias reals to a model of GCH. All of this should be in Blass' chapter of the Handbook. – Miha Habič May 8 '15 at 0:57
• @MihaHabič: Exactly! I wrote in the post that I knew that there were no relations between $\mathfrak{s}$ and $\text{cov}(\mathcal{M})$ provable in ZFC, but I had had problems with finding references describing appropriate models. I will look into the Handbook, thank you. But I am still interested in the more general question asked in the post. – Damian Sobota May 8 '15 at 1:10
• Oh, I'm sorry, I didn't read your question closely enough. As far as I know controlling three characteristics at once is still quite difficult. I have heard Diego Mejia speak on the topic and he seems to have a quite general construction for separating $\mathfrak{s}, \mathrm{cov}(\mathcal{M})$ and $\mathfrak{c}$ (although $\mathfrak{b}=\mathfrak{s}$ in his model). See Theorem 4.4 of here. – Miha Habič May 8 '15 at 1:44
• And the 'Handbook' is? - EDIT: ah, you mean math.lsa.umich.edu/~ablass/hbk.pdf – David Roberts May 8 '15 at 1:55

In Diego Mejía's reference (mentioned by Habic)

http://arxiv.org/abs/1305.4739

there are models for triples $(\kappa,\lambda,\mu)$ for

1) $\mathfrak{s}=\kappa<\mathfrak{b}=\mathrm{cov}(\mathcal{M})=\lambda<\mathfrak{c}=\mu$,

2) $\mathfrak{s}=\mathfrak{b}=\kappa<\mathrm{cov}(\mathcal{M})=\lambda<\mathfrak{c}=\mu$.

These models are constructed by finite support iterations (fsi) of ccc forcings where $\kappa<\lambda$ are regular and $\lambda<\mu=\mu^{<\kappa}$.

There are other two models constructed by fsi of ccc posets:

3) $\mathfrak{b}=\kappa<\mathfrak{s}=\mathrm{cov}(\mathcal{M})=\lambda<\mathfrak{c}=\mu$.

Use Brendle-Fischer matrix iteration construction for $\mathfrak{b}=\mathfrak{a}=\kappa<\mathfrak{s}=\lambda$, but change the length of the iteration to $\mu\lambda$ (ordinal product), see Section 4 in

http://projecteuclid.org/euclid.jsl/1294170995

4) $\mathfrak{s}=\mathrm{cov}(\mathcal{M})=\aleph_1<\mathfrak{b}=\kappa<\mathfrak{c}=\lambda$.

Start with a model of $\mathfrak{b}=\mathfrak{c}=\kappa$ and force with the random algebra to add $\mu$-many random reals side-by-side. This preserves the value of $\mathfrak{b}$, makes $\mathrm{non}(\mathcal{N})=\aleph_1$ and $\mathrm{cov}(\mathcal{N})=\mathfrak{c}=\mu$.

• Thank you very much. I will analyse all cases and if I have a question, I will ask. However in fact, the paper of Diego is sufficient for me. Regards! – Damian Sobota May 11 '15 at 22:43