3. Problems in the Overlap of High and Low Forcing

Problem 3.1.
Is there a model for $\text{TP}_{\kappa^{++}}$, with $\kappa$ a singular strong limit of countable cofinality, where we use side conditions in place of the Mitchell poset? 
Problem 3.2.
Is there a model for $\text{TP}_{\aleph_2} + \text{TP}_{\aleph_3}$ using side conditions? 
Problem 3.3.
Is there a model where both $I[\aleph_2]$ and $I[\aleph_3]$ are trivial? In other words, is $\text{AP}_{\aleph_2} + \text{AP}_{\aleph_3}$ consistent? 
Problem 3.4.
Is the forcing axiom for proper posets of size $\aleph_1$ with $\aleph_2$many dense sets consistent? If so, what if we replace the size$\aleph_2$ requirement with the $\aleph_2$chain condition? 
There is a coloring $F:[\aleph_2]^2 \rightarrow {0,1}$ of pairs from $\aleph_2$ in $2$ colors so that the poset of finite approximations to a $0$ or $1$homogeneous set has the $\aleph_2$chain condition. (We mean either the poset $\mathbb P$ of finite functions $f:\omega \rightarrow F^{1}(0)$ or the poset of $f:\omega \rightarrow F^{1}(0)$, ordered by inclusion)
Problem 3.5.
Can this forcing be made proper using side conditions? Does this preserve the $\aleph_2$chain condition? 
Problem 3.6.
[David Aspero] Suppose $\aleph_\omega$ is a strong limit. Is there a poset ${\mathbb P}$ of size $<\aleph_\omega$, which adds a club guessing sequence for $S^{\omega_2}_{\omega_1}$, preserving $\omega_1$, $\omega_2$, and $\omega_3$.
Cite this as: AimPL: High and Low forcing, available at http://aimpl.org/highlowforcing.