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A dominating family on $\omega^\omega$ is a set $\mathcal D \subset \omega^\omega$ such that for every $f \in \omega^\omega$ there exists $g \in \mathcal D$ such that $f<^* g$ (that is, $f(n)<g(n)$ for all but a finite number of $n$'s). The smallest cardinality of a dominating family is $\mathfrak d$, and it is well known that $\omega_1\leq \mathfrak d\leq \mathfrak c$ and that it is consistent that $\mathfrak d$ may be almost everything between $\omega_1$ and $\mathfrak c$. I know that this is an important combinatorical concept and that it is widely used/studied.

I am interested in reading about some related questions. One may also define dominating families on $\omega^{\omega_1}$ and on $\omega_1^{\omega_1}$, and then define $\mathfrak d_{\omega_1, \omega}$ and $\mathfrak d_{\omega_1, \omega_1}$ in an analogous manner (so $\mathfrak d_{\omega, \omega}=\mathfrak d$).

I would like to know more about dominating families of higher cardinalities, so my main question is "where may I learn more about them?" Can someone please tell me some references?

Some questions (out of my mind) I would like to explore are:

  1. It is easy to see that $\mathfrak d\leq \mathfrak d_{\omega_1, \omega}$. However, are they the same? What are the possible relations between them?
  2. Since $\omega_1$ is regular, it is easy to see that $\omega_2\leq \mathfrak d_{\omega_1, \omega_1}\leq 2^{\omega_1}$. Is it consistent that $\mathfrak d_{\omega_1, \omega_1}<2^{\omega_1}$?
  3. Is it consistent that $\mathfrak d_{\omega_1, \omega}=\omega_1$?
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3 Answers 3

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Here is some general background information. The relevant search phrases for this topic are generalized cardinal invariants or generalized cardinal characteristics, and the topic has a growing literature, emerging over many years. The topic has been studied as folklore for some time.

Here are a few specific resources:

What you call $\mathfrak{d}_{\omega_1,\omega_1}$ is known as $\mathfrak{d}_{\omega_1}$. Your concept of $\mathfrak{d}_{\omega_1,\omega}$ is less studied.

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  • $\begingroup$ I think that there are many additional references, and perhaps others can mention them. $\endgroup$
    – Joel David Hamkins
    Commented Feb 11, 2018 at 21:08
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Professor Hamkins already gave many interesting references. Let me add a few more.

Possibly, the work of Cummings-shelah Cardinal invariants above the continuum is the starting point for the study of generalizations of cardinal invariants to the context of uncountable cardinals. In this paper, they prove the following: If $λ↦(β(λ),δ(λ),μ(λ))$ is a class function from regular cardinals into the cube of cardinals satisfying $λ^+≤β(λ)=cf(β(λ))≤cf(δ(λ))≤δ(λ)≤μ(λ)$ and $cf(μ(λ))>λ$ for all $λ$, then there exists a model where $b(λ)=β(λ), \mathfrak{d}(λ)=δ(λ)$, and $2^λ=μ(λ)$ for all $λ$.

for some other references see

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Two more references: Monk,J.D. Notre Dame J. Formal Logic 45 (2004),129-146 Szymanski, A. Proc. AMS 104 (1988), 596-602.

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