Almost Disjoint Transversals

Definition: Suppose that $\kappa$ and $\mu$ are cardinals with $\kappa<\mu$, and $J$ is an ideal on $\kappa$. We say that $\mathcal{F}\subseteq {}^\kappa\mu$ is a $J$-adt (almost disjoint transversal) if for every $f\neq g\in \mathcal{F}$, we have that $f\neq_J g$.

One reason for looking at almost disjoint transversals is that they show up in the context of cardinal arithmetic quite often, as they're used to prove the first Galvin-Hajnal formula and they appear in several of Shelah's papers (e.g. 186, 410, 506, and 589). In particular, we want to consider the following cardinal:

$T_J(\mu)=\sup\{|F| : F\subseteq{}^\kappa\mu\text{ is a }J\text{-adt}\}$

I want to use the next few posts to work through a result of Shelah that, under reasonable assumptions on the ideal $J$, relates $T_J(\mu)$ to a pcf-theoretic cardinal. This is all based on the arguments in section 6 of Shelah paper 410. In the remainder of this post we will define some auxiliary invariants, with the goal of showing all of these are equal in most cases. First, we need some definitions.

Definition: Given an ideal $J$ on a set $A$, we say that $A$ is $J$-positive ($A\in J^+$) if $A$ is not in $J$.

Definition: Given a cardinal $\theta$, we say that an ideal $J$ is $\theta$-based if for any $J$-positive $A$, there is a $J$-positive $B\subseteq A$ such that $|B|<\theta$.

At this point, we will assume that our ideal $J$ is $\theta$-based where $\kappa^{<\theta}<\mu$. At the end of this series of posts, we will discuss some situations under which these assumptions are satisfied, as they are not terribly stringent. Finally, we need one more definition which is a bit ad-hoc, but will help us in defining the invariants I mentioned earlier.

Definition: We say that a cardinal $\lambda$ is weakly $J$-representable if there is a block sequence $\langle E(i) : i<\kappa\rangle$ with $E(i)\in[(\kappa^{<\sigma,}\mu]\cap\mathrm{Reg}]^{<\omega}$ such that for any $J$-positive set $A$, $\lambda\leq\max pcf\bigcup_{i\in A}E(i)$. We say that $\lambda$ is $J$-representable if instead we ask $\lambda=\max pcf\bigcup_{i\in A}E(i)$ for every $J$-positive set $A$.

$T^2_J(\mu)=\sup\{\lambda : \lambda\text{ is weakly }J\text{-representable}\}$
$T^3_J(\mu)=\sup\{\lambda : \lambda\text{ is }J\text{-representable}\}$

Note here that the above cardinals are defined in terms of a $\kappa$-sequence of finite blocks, which may seem somewhat strange initially. It's worth pointing out that this "chunky pcf" appears in a number of other places in Shelah's work primarily when Skolem Hulls and towers of elementary submnodels are involved. In a later post, we will see how the chunkyness of the pcf gets used. In the next post however, I will discuss how $T_J(\mu)$, $T^2_J(\mu)$, and $T^3_J(\mu)$ are related and define one last related invariant.