\DeclareMathOperator{\pp}{pp}

\DeclareMathOperator{\tcf}{tcf}

\DeclareMathOperator{\pcf}{pcf}

\DeclareMathOperator{\cov}{cov}

\def\cf{\rm{cf}}

\DeclareMathOperator{\otp}{otp}

\def\REG{\sf {REG}}

\def\restr{\upharpoonright}

\def\bd{\rm{bd}}

\def\subs{\subseteq}

\def\cof{\rm{cof}}

\def\ran{\rm{ran}}

\DeclareMathOperator{\ch}{Ch}

\DeclareMathOperator{\PP}{PP}

\DeclareMathOperator{\Sk}{Sk}

\DeclareMathOperator{\supp}{supp}

\)

**Definition of $P_\alpha$**

**Given $\alpha<\kappa$, we define the partial order $P_\alpha$ to be those conditions in $P_\kappa$ that satisfy**

- $w_p\subseteq\alpha$, and
- $N\cap m_p(N)\subseteq\alpha$ for all $N\in\mathcal{N}_p$

The last condition says that the implicit coordinates arising from $N$ are all less than $\alpha$. This looks different than what Aspero-Mota do because we're requiring that markers are actually elements of the model. This is probably only a cosmetic difference; on the face of it, conditions with the property we demand should be dense in the Aspero-Mota poset.

**Restrictions of conditions**

Give a condition $p\in P_\kappa$ and $\alpha<\kappa$, we define $p\restr\alpha$ by setting

- $\mathcal{N}_{p\restr\alpha}=\mathcal{N}_p$
- for $N\in\mathcal{N}_{p\restr\alpha}$ we define $m_{p\restr\alpha}(N)=\min\{m_p(N),\min(N\setminus\alpha)\}$
- $f_{p\restr\alpha}= f_p\restr \alpha$

**Properties of Restrictions of Conditions**

*Claim 1: If $p\in P_\kappa$ and $\alpha<\kappa$ then $p\restr\alpha$ is a condition in $P_\alpha$.*

First check that $p\restr\alpha$ is actually a condition in $P_\kappa$, and the rest follows immediately from the definition of $p\restr\alpha$. $_\square$

*Claim 2: $p$ is an extension of $p\restr\alpha$ in $P_\kappa$.*

*Claim 3: If $q\leq p\restr\alpha$ in $P_\alpha$ then $q$ is compatible with $p$.*

*We define $r=\langle \mathcal{N}_r, m_r, f_r\rangle$ by setting*

- $f_r=f_p\cup f_q$
- $\mathcal{N}_r = \mathcal{N}_q$
- $m_r(N)=m_p(N)$ for $N\in\mathcal{N}_p$, and $m_r(N)=m_q(N)$ otherwise.

We must check that $r$ is a condition in $P_\kappa$ extending both $q$ and $p$.

Clearly $\mathcal{N}_r$, $m_r$, and $f_r$ all have the right form, so we need to check first that $f_r$ is consistent with the demands imposed by $\mathcal{N}_r$ and $m_r$. This is immediate for coordinates in $w_r$ below $\alpha$, as $f_r$ and $f_q$ agree. For coordinates at or above $\alpha$, we know that $f_r$ agrees with $f_p$ and so is consistent with the models in $\mathcal{N}_p$. The new models that appear in $\mathcal{N}_q$ all have markers at most $\alpha$, and so impose no restrictions on these coordinates.

Note that $r$ extends $q$ (we've potentially added new coordinates, and increased the markers on models from $\mathcal{N}_p$ back to their original values), and also extends $p$ (as $\mathcal{N}_r$ and $f_r$ extend $\mathcal{N}_p$ and $f_p$, while the marker function has remained the same on models from $\mathcal{N}_p$.)

NB. This is a place where allowing models of the same height to have unrelated markers might be relevant in the more general construction

*Claim 4: $P_\alpha$ is a regular suborder of $P_\kappa$.*

If $\mathcal{A}$ is a maximal antichain in $P_\alpha$ and $p\in P_\kappa$, then $p$ is compatible with an element of $\mathcal{A}$ using the preceding argument, as we simply let $q$ be an extension of $p\restr\alpha$ in $\mathcal{A}$..