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1. Problems in Low Forcing

"Low Forcing" refers to the method of side conditions, specifically where the side conditions are elementary submodels of some large transitive structure.
    1.     Let $\mathbb{C} = (C,\le)$ be a linear order of size $\kappa$ and consider its Cartesian square under the ordering $(x_1,y_1) \le (x_2,y_2)$ iff $x_1 \le x_2$ and $y_1 \le y_2$. $\mathbb C$ is a Countryman line if this Cartesian square is the union of less than $\kappa$-many chains (i.e. linearly ordered subsets).

      Problem 1.05.

      [Justin Moore] Is it consistent (with the continuum hypothesis) that there is a minimal $\aleph_2$-Countryman line—that is, an $\aleph_2$-Countryman line that order-embeds into all others?
        •     Two linear orders $\mathbb{C}_1$ and $\mathbb{C}_2$ are near if there is another linear order $\mathbb{C}_0$ that embeds into both of them. $\mathbb{C}_1$ and $\mathbb{C}_2$ are co-near if there is a linear order embedding into $\mathbb{C}_1$ and $\mathbb{C}_2^\ast$, where $\mathbb{C}_2^\ast$ is the reverse of $\mathbb{C}_2$.

          Problem 1.1.

          [Justin Moore] Is it consistent with the continuum hypothesis that any two $\aleph_2$-Countryman lines are near or co-near?
            •     Strong homology is a homology theory satisfying the Eilenberg-Steenrod axioms and is invariant under strong shape.

              Problem 1.15.

              [Justin Moore] Is strong homology consistently additive for closed subspaces of Euclidean space, or more generally for locally compact metric spaces?
                  CH and PFA both imply a negative answer to this question. It is likely that a positive answer to this question implies that $\mathfrak{d}<\aleph_\omega$, where $\mathfrak{d}$ is the dominating number of the continuum.
                •     $\mathbb A = \langle A_f : f \in \omega^\omega \rangle$ is the following inverse system of abelian groups:

                  $$ A_f = \bigoplus_{n \in \omega} \bigoplus_{i < f(n)} \mathbb Z = \bigoplus_{n \in \omega} {\mathbb Z}^{f(n)}$$

                  Let ${\mathbb Z}^{\omega \times \omega}/fin$ be ${\mathbb Z}^{\omega \times \omega}$ modulo finite equivalence, let $G_f = \prod_{n \in \omega} {\mathbb Z}^{f(n)}/fin$, and consider the chain complex

                  $${\mathbb Z}^{\omega \times \omega}/fin \xrightarrow{\delta} \prod_{f_0 \in \omega^\omega} G_{f_0} \xrightarrow{\delta} \prod_{f_0 \le f_1} G_{f_0} \xrightarrow{\delta} \prod_{f_0 \le f_1 \le f_2} G_{f_0} \xrightarrow{\delta} \ldots$$


                  $$\delta_s(f_0,f_1,\ldots,f_n) = \sum^n_{i = 0}(-1)^i s(f_0,\ldots,\hat{f}_i,\ldots,f_n)$$

                  for $s \in \prod_{f_0 \le \ldots \le f_{n-1}} G_{f_0}$.

                  We define $\lim^p \mathbb A \cong \ker(\delta^p)/\text{im}(\delta^{p-1})$.

                  Problem 1.2.

                  [Justin Moore] Is it consistent to have $\lim^p \mathbb{A} = 0$ for all $p$?
                      If strong homology is consistent for locally compact metric spaces, then the answer to this question is positive.
                    • Problem 1.25.

                      Is it consistent that there is a poset which adds a club in $\omega_3$ which contains no infinite ground model subset over a model of GCH?
                        • Problem 1.3.

                          Is there a large cardinal hypothesis that proves the bounded forcing axiom for Namba forcing?
                            • Problem 1.35.

                              Can Namba forcing be iterated with side conditions? Does this work if we replace Namba forcing with a forcing satisfying Shelah’s $S$-condition?
                                • Problem 1.4.

                                  Assume CH. Is there a strongest forcing axiom of $\sigma$-closed posets meeting $\aleph_2$-many dense sets? What if we do not assume CH?
                                    • Conjecture 1.45.

                                      The following is inconsistent: “For every $\Sigma_1$ statement $\varphi$ with $\aleph_1$ and $\aleph_2$ as parameters, if $\varphi$ can be forced by an $\aleph_1, \aleph_2$-preserving forcing, then $\varphi$ holds.”
                                        •     If $X$ is a topological space, let $s(X)=\sup\{|Y|:Y \text{ is a discrete subspace of }X\}$. It is a theorem that $|X| \le 2^{2^{s(X)}}$.

                                          Problem 1.5.

                                          [Stevo Todorčević] When can one obtain $|X|\le 2^{s(X)}$?
                                              The answer is negative under $\Diamond$, even for compact Hausdorff spaces. PFA implies that the answer is negative for $X$ with $s(X)=\aleph_0$. PFA also implies that $|X| \le 2^{\aleph_0}$ for every Hausdorff space with no uncountable discrete subspaces.
                                            • Problem 1.55.

                                              [Stevo Todorčević] Do any of the analogs of PFA at $\aleph_2$ give us $|X| \le 2^{\aleph_1}$ for every Hausdorff space with no discrete subspaces of size $\aleph_2$?
                                                •     It is a fact that a compact Hausdorff space $X$ is countably tight if and only if $X$ contains no $\omega_1$-free sequence.

                                                  Problem 1.6.

                                                  [Stevo Todorčević] Suppose $X$ is a compact Hausdorff space of tightness $\aleph_1$ (i.e. there are no $\omega_2$-free sequences) and density $\aleph_1$. Is it true that $|X| \le 2^{\aleph_1}$?
                                                    • Problem 1.65.

                                                      [Stevo Todorčević] Under a given analog of PFA for $\aleph_2$, how many cofinal types of directed sets of cardinality $\aleph_2$ are there?
                                                          Under PFA, $1, \omega, \omega_1, \omega \times \omega_1$, and $(\omega_1)^{<\omega}$ are the cofinal types of directed sets of cardinality $\aleph_1$.
                                                        •     $\kappa \rightarrow (\alpha,\beta)^n$ means that for every coloring $F:[\kappa]^n \rightarrow \{0,1\}$, then then there is either a homogeneous set $H_0$ of order-type $\alpha$ such that $F"H_0=\{0\}$, or else a homogeneous set $H_1$ such that $F"H_1=\{1\}$.

                                                          Problem 1.7.

                                                          [Stevo Todorčević] Can $\omega_2 \rightarrow (\omega_2,\alpha)^2$ be deduced from one of the forcing axioms at $\aleph_2$.
                                                              It is known that PFA implies $\omega_1 \rightarrow (\omega_1,\alpha)^2$ for all $\alpha<\omega_1$.
                                                            •     It is known under PFA that the gaps-spectrum of $P(\omega)/fin$ consists of $(\omega_1,\omega_1^\ast),(\omega_2,\omega^\ast)$, and $(\omega,\omega_2^\ast)$.

                                                              Problem 1.75.

                                                              [Stevo Todorčević] Assuming an appropriate generalization of PFA, determine the gaps spectrum of $P(\omega)/fin$.

                                                                  Cite this as: AimPL: High and Low forcing, available at http://aimpl.org/highlowforcing.