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3. Problems in the Overlap of High and Low Forcing

    1. 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?
                          If we do obtain properness, then the fprcing axiom for \aleph_2-many dense sets proper posets of size \aleph_1—and for proper posets with the \aleph_2-chain condition—is inconsistent.
                        • 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.
                              If yes than by PCF theory, in ZFC it would be provable that 2^{\aleph_\omega}<\aleph_{\omega_3} when \aleph_\omega is a strong limit.

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