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1. Sealing from iterability

   Sargsyan, Grigor; Trang, Nam (January 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   We show that if V has a proper class ofWoodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy (as defined in the paper) then Sealing holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to other work by the authors where it is shown that Sealing holds in a generic extension of a certain minimal universe. The current theorem is more general in that no minimality assumption is needed. A corollary of the main theorem is that Sealing is consistent relative to the existence of a Woodin cardinal which is a limit of Woodin cardinals. This improves significantly on the first consistency of Sealing obtained by W.H. Woodin. The Largest Suslin Axiom (LSA) is a determinacy axiom isolated byWoodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let LSA-over-uB be the statement that in all (set) generic extensions there is a model of LSA whose Suslin, co-Suslin sets are the universally Baire sets. The other main result of the paper shows that assuming V has a proper class of inaccessible cardinals which are limit of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, in the universe V [g], where g is V -generic for the collapse of the successor of the least strong cardinal to be countable, the theory LSA-over-UB fails; this implies that LSA-over-UB is not equivalent to Sealing (over the base theory of V [g]). This is interesting and somewhat unexpected, in light of other work by the authors. Compare this result with Steel’s well-known theorem that “AD in L(R) holds in all generic extensions” is equivalent to “the theory of L(R) is sealed” in the presence of a proper class of measurable cardinals. 

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2. DIVERGENT MODELS WITH THE FAILURE OF THE CONTINUUM HYPOTHESIS

   https://doi.org/10.1017/jsl.2023.92  

   TRANG, NAM (December 2023, The Journal of Symbolic Logic)

   Abstract We construct divergent models of$$\mathsf {AD}^+$$along with the failure of the Continuum Hypothesis ($$\mathsf {CH}$$) under various assumptions. Divergent models of$$\mathsf {AD}^+$$play an important role in descriptive inner model theory; all known analyses of HOD in$$\mathsf {AD}^+$$models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of$$\mathsf {AD}^+$$. Our results are the first step toward resolving various open questions concerning the length of definable prewellorderings of the reals and principles implying$$\neg \mathsf {CH}$$, like$$\mathsf {MM}$$, that divergent models shed light on, see Question 5.1. 

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3. Almost Everywhere Behavior of Functions According to Partition Measures

   https://doi.org/10.1017/fms.2023.130  

   Chan, William; Jackson, Stephen; Trang, Nam (January 2024, Forum of Mathematics, Sigma)

   Abstract This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals. The following summarizes the main results proved under suitable partition hypotheses.•If$$\kappa $$is a cardinal,$$\epsilon < \kappa $$,$${\mathrm {cof}}(\epsilon ) = \omega $$,$$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$$and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the almost everywhere short length continuity property: There is a club$$C \subseteq \kappa $$and a$$\delta < \epsilon $$so that for all$$f,g \in [C]^\epsilon _*$$, if$$f \upharpoonright \delta = g \upharpoonright \delta $$and$$\sup (f) = \sup (g)$$, then$$\Phi (f) = \Phi (g)$$.•If$$\kappa $$is a cardinal,$$\epsilon $$is countable,$$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$$holds and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the strong almost everywhere short length continuity property: There is a club$$C \subseteq \kappa $$and finitely many ordinals$$\delta _0, ..., \delta _k \leq \epsilon $$so that for all$$f,g \in [C]^\epsilon _*$$, if for all$$0 \leq i \leq k$$,$$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$$, then$$\Phi (f) = \Phi (g)$$.•If$$\kappa $$satisfies$$\kappa \rightarrow _* (\kappa )^\kappa _2$$,$$\epsilon \leq \kappa $$and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the almost everywhere monotonicity property: There is a club$$C \subseteq \kappa $$so that for all$$f,g \in [C]^\epsilon _*$$, if for all$$\alpha < \epsilon $$,$$f(\alpha ) \leq g(\alpha )$$, then$$\Phi (f) \leq \Phi (g)$$.•Suppose dependent choice ($$\mathsf {DC}$$),$${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$$and the almost everywhere short length club uniformization principle for$${\omega _1}$$hold. Then every function$$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$$satisfies a finite continuity property with respect to closure points: Let$$\mathfrak {C}_f$$be the club of$$\alpha < {\omega _1}$$so that$$\sup (f \upharpoonright \alpha ) = \alpha $$. There is a club$$C \subseteq {\omega _1}$$and finitely many functions$$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$$so that for all$$f \in [C]^{\omega _1}_*$$, for all$$g \in [C]^{\omega _1}_*$$, if$$\mathfrak {C}_g = \mathfrak {C}_f$$and for all$$i < n$$,$$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$$, then$$\Phi (g) = \Phi (f)$$.•Suppose$$\kappa $$satisfies$$\kappa \rightarrow _* (\kappa )^\epsilon _2$$for all$$\epsilon < \kappa $$. For all$$\chi < \kappa $$,$$[\kappa ]^{<\kappa }$$does not inject into$${}^\chi \mathrm {ON}$$, the class of$$\chi $$-length sequences of ordinals, and therefore,$$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$$. As a consequence, under the axiom of determinacy$$(\mathsf {AD})$$, these two cardinality results hold when$$\kappa $$is one of the following weak or strong partition cardinals of determinacy:$${\omega _1}$$,$$\omega _2$$,$$\boldsymbol {\delta }_n^1$$(for all$$1 \leq n < \omega $$) and$$\boldsymbol {\delta }^2_1$$(assuming in addition$$\mathsf {DC}_{\mathbb {R}}$$). 

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4. THE TREE PIGEONHOLE PRINCIPLE IN THE WEIHRAUCH DEGREES

   https://doi.org/10.1017/jsl.2025.11  

   DZHAFAROV, DAMIR D; SOLOMON, REED; VALENTI, MANLIO (February 2025, The Journal of Symbolic Logic)

   Abstract We study versions of the tree pigeonhole principle,$$\mathsf {TT}^1$$, in the context of Weihrauch-style computable analysis. The principle has previously been the subject of extensive research in reverse mathematics, an outstanding question of which investigation is whether$$\mathsf {TT}^1$$is$$\Pi ^1_1$$-conservative over the ordinary pigeonhole principle,$$\mathsf {RT}^1$$. Using the recently introduced notion of the first-order part of an instance-solution problem, we formulate the analog of this question for Weihrauch reducibility, and give an affirmative answer. In combination with other results, we use this to show that unlike$$\mathsf {RT}^1$$, the problem$$\mathsf {TT}^1$$is not Weihrauch requivalent to any first-order problem. Our proofs develop new combinatorial machinery for constructing and understanding solutions to instances of$$\mathsf {TT}^1$$. 

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5. On the weight zero compactly supported cohomology of H_{g,n}

   https://doi.org/10.1017/fms.2024.53  

   Brandt, Madeline; Chan, Melody; Kannan, Siddarth (January 2024, Forum of Mathematics, Sigma)

   Abstract For$$g\ge 2$$and$$n\ge 0$$, let$$\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$$denote the complex moduli stack ofn-marked smooth hyperelliptic curves of genusg. A normal crossings compactification of this space is provided by the theory of pointed admissible$$\mathbb {Z}/2\mathbb {Z}$$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of$$\mathcal {H}_{g, n}$$. Using this graph complex, we give a sum-over-graphs formula for the$$S_n$$-equivariant weight zero compactly supported Euler characteristic of$$\mathcal {H}_{g, n}$$. This formula allows for the computer-aided calculation, for each$$g\le 7$$, of the generating function$$\mathsf {h}_g$$for these equivariant Euler characteristics for alln. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissibleG-covers of genus zero curves, whenGis abelian, as a symmetric$$\Delta $$-complex. We use these complexes to generalize our formula for$$\mathsf {h}_g$$to moduli spaces ofn-pointed smooth abelian covers of genus zero curves. 

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