Dominating families in bigger cardinals 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:


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*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?

*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 $\mathfrak d_{\omega_1, \omega_1}<2^{\omega_1}$?

*Is it consistent that $\mathfrak d_{\omega_1, \omega}=\omega_1$?

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


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*Dilip Raghavan, Saharon Shelah, Two results on cardinal invariants at uncountable cardinals, arxiv.org:1801.09369.

*Brendle, Jörg, Cardinal invariants of the continuum and combinatorics on uncountable cardinals, Ann. Pure Appl. Logic 144, No. 1-3, 43-72 (2006). ZBL1112.03046.

*Cichon's Diagram for uncountable cardinals, 
Joerg Brendle, Andrew Brooke-Taylor, Sy-David Friedman, Diana Montoya
arxiv:1611.08140. 

*Brooke-Taylor, A.D.; Fischer, V.; Friedman, S.D.; Montoya, D.C., Cardinal characteristics at $\kappa$ in a small $\mathfrak{u}(\kappa)$ model,  ZBL06643764. 

*Slides for a nice talk by Luke Serafin, Cardinal invariants of the generalized continua. 
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. 
A: 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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*Cardinal Invariants $\mathbf{b}_κ$ and $\mathbf{t}_κ$.

*Generalized Domination.

*Two inequalities between cardinal invariants

*Adding dominating functions mod finite
