If kappa is an infinite cardinal, the boldface GCH at kappa is the statement that kappa(+) does not inject into P(kappa). It will be shown here that omega(1) -> (omega(1))(2)(omega 1) (the strong partition property at w1) and j(mu 1 omega 1) (omega(1)) = omega(2) (the ultrapower of w(1) by the club filter on omega(1) is omega(2)) imply that the boldface GCH holds at wn for all n < w using combinatorial arguments. In particular, AD implies the boldface GCH holds at omega(n) for all n < omega. (c) 2026 The Author(s). Published by Elsevier B.V.

Within the determinacy setting, P(omega(1)) is regular (in the sense of cofinality) with respect to many known cardinalities and thus there is substantial evidence to support the conjecture that P(omega(1)) has globally regular cardinality. However, there is no known information about the regularity of P(omega(2)) . It is not known if P(omega(2)) is even 2 -regular under any determinacy assumptions. The article will provide the following evidence that P(omega(2)) may possibly be omega(1)-regular: Assume AD(+) . If (A(alpha):alpha

Woodin introduced an extension of the axiom of determinacy, AD, called AD+ which includes an assertion that all sets of reals have an infinity-Borel code. An infinity-Borel code is a pair (phi,S) where phi is a formula and S is a set of ordinals which provides a highly absolute definition for a set of reals. This paper will use AD+ and infinity-Borel codes to establish a property of ordinal definability analogous to a property for Sigma 11 shown by Harrington-Shore-Slaman (2017). Under AD+, the paper will also use infinity-Borel codes to explore the cardinality of sets below P(omega 1) which Woodin (2006) began investigating under ADR and DC. The following summarizes the main results. Assume ZF+AD++V=L(P(R)). If H subset of R has the property that there is a nonempty OD set of reals K such that H is ODz for any z is an element of K, then H is OD. Assume ZF+AD++ADR+V=L(P(R)). Then there is a cardinal strictly between |[omega 1]ADR+V=L(P(R)).

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 is a cardinal, , , and , then satisfies the almost everywhere short length continuity property: There is a club and a so that for all , if and , then . If is a cardinal, is countable, holds and , then satisfies the strong almost everywhere short length continuity property: There is a club and finitely many ordinals so that for all , if for all , , then . If satisfies , and , then satisfies the almost everywhere monotonicity property: There is a club so that for all , if for all , , then . Suppose dependent choice ( ), and the almost everywhere short length club uniformization principle for hold. Then every function satisfies a finite continuity property with respect to closure points: Let be the club of so that . There is a club and finitely many functions so that for all , for all , if and for all , , then . Suppose satisfies for all . For all , does not inject into , the class of -length sequences of ordinals, and therefore, . As a consequence, under the axiom of determinacy , these two cardinality results hold when is one of the following weak or strong partition cardinals of determinacy: , , (for all ) and (assuming in addition ).

For each cardinal kappa, let B(kappa) be the ideal of bounded subsets of kappa and P-kappa(kappa) be the ideal of subsets of kappa of cardinality less than kappa. Under determinacy hypothesis, this paper will completely characterize for which cardinals kappa there is a nontrivial maximal B(kappa) almost disjoint family. Also, the paper will completely characterize for which cardinals kappa there is a nontrivial maximal P-kappa(kappa) almost disjoint family when kappa is not an uncountable cardinal of countable cofinality. More precisely, the following will be shown.Assuming AD(+), for all kappa < Theta, there are no maximal B(kappa) almost disjoint families A such that (|A| < cof(kappa)). For all kappa < Theta, if cof(kappa) > omega, then there are no maximal P-kappa(kappa) almost disjoint families A so that (|A|(|A|= Theta. For any cardinal kappa with cof(kappa) > omega, there is a maximal P-kappa(kappa) almost disjoint family if and only if cof(kappa) >= Theta.(c) 2023 Elsevier Inc. All rights reserved.

RCA0 is a weak subsystem of second order arithmetic which will serve as the base theory. WKL0 is the subsystem of second order arithmetic corresponding to the weak König lemma which states that every infinite tree on {0,1} has an infinite path. (Σ10∧Π10,Δ10)-LDET2 is the principle asserting that one of the two players has a winning strategy in all Lipschitz game with moves from {0,1} such that Player 1 owns a set which is the intersection of an open and a closed subset of 2N in its natural topology and Player 2 owns a clopen subset of 2N. It will be shown that RCA0 proves that WKL0 is equivalent to (Σ10∧Π10,Δ10)-LDET2.

Abstract Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $ , $\vee $ , $\forall ^{\mathbb {R}}$ , continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $ . Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $ . Then the countable length everywhere club uniformization holds for $\kappa $ : For every relation $R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$ with the property that for all $\ell \in {}^{<{\omega _1}}\kappa $ and clubs $C \subseteq D \subseteq \kappa $ , $R(\ell ,D)$ implies $R(\ell ,C)$ , there is a uniformization function $\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$ with the property that for all $\ell \in \mathrm {dom}(R)$ , $R(\ell ,\Lambda (\ell ))$ . In particular, under these assumptions, for all $n \in \omega $ , $\boldsymbol {\delta }^1_{2n + 1}$ satisfies the countable length everywhere club uniformization.

Assume [Formula: see text]. If [Formula: see text] is an ordinal and X is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every [Formula: see text] and function [Formula: see text], there is a club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. For every [Formula: see text] and function [Formula: see text], there is an [Formula: see text]-club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of [Formula: see text]. [Formula: see text]. [Formula: see text]. [Formula: see text]. It will also be shown that [Formula: see text] has the Jónsson property: For every [Formula: see text], there is an [Formula: see text] with [Formula: see text] so that [Formula: see text].

Assume Z F + A D + D C R \mathsf {ZF} + \mathsf {AD} + \mathsf {DC}_\mathbb {R} . There is no injection of > ω 1 ω 1 {}^{>\omega _{1}}{\omega _{1}} (the set of countable length sequences of countable ordinals) into ω O N {}^\omega \mathrm {ON} (the class of ω \omega length sequences of ordinals). There is no injection of [ ω 1 ] ω 1 [{\omega _{1}}]^{{\omega _{1}}} (the powerset of ω 1 {\omega _{1}} ) into > ω 1 O N {}^{>{\omega _{1}}}\mathrm {ON} (the class of countable length sequences of ordinals).

Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. Let $\approx$ denote the relation of being in bijection. Let $\kappa \in \mathrm{ON}$ and $\langle E_\alpha : \alpha < \kappa\rangle$ be a sequence of equivalence relations on $\mathbb{R}$ with all classes countable and for all $\alpha < \kappa$, $\mathbb{R} / E_\alpha \approx \mathbb{R}$. Then the disjoint union $\bigsqcup_{\alpha < \kappa} \mathbb{R} / E_\alpha$ is in bijection with $\mathbb{R} \times \kappa$ and $\bigsqcup_{\alpha < \kappa} \mathbb{R} / E_\alpha$ has the J\'onsson property. Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. A set $X \subseteq [\omega_1]^{<\omega_1}$ has a sequence $\langle E_\alpha : \alpha < \omega_1\rangle$ of equivalence relations on $\mathbb{R}$ such that $\mathbb{R} / E_\alpha \approx \mathbb{R}$ and $X \approx \bigsqcup_{\alpha < \omega_1} \mathbb{R} / E_\alpha$ if and only if $\mathbb{R} \sqcup \omega_1$ injects into $X$. Assume $\mathsf{AD}$. Suppose $R \subseteq [\omega_1]^\omega \times \mathbb{R}$ is a relation such that for all $f \in [\omega_1]^\omega$, $R_f = \{x \in \mathbb{R} : R(f,x)\}$ is nonempty and countable. Then there is an uncountable $X \subseteq \omega_1$ and function $\Phi : [X]^\omega \rightarrow \mathbb{R}$ which uniformizes $R$ on $[X]^\omega$: that is, for all $f \in [X]^\omega$, $R(f,\Phi(f))$. Under $\mathsf{AD}$, if $\kappa$ is an ordinal and $\langle E_\alpha : \alpha < \kappa\rangle$ is a sequence of equivalence relations on $\mathbb{R}$ with all classes countable, then $[\omega_1]^\omega$ does not inject into $\bigsqcup_{\alpha < \kappa} \mathbb{R} / E_\alpha$.

ZF + AD proves that for all nontrivial forcings $$\mathbb{P}$$ on a wellorderable set of cardinality less than Θ, $${1_\mathbb{P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}$$ . ZF + AD + Θ is regular proves that for all nontrivial forcing $$\mathbb{P}$$ which is a surjective image of ℝ, $${1_\mathbb{P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}$$ . In particular, ZF + AD + V = L(ℝ) proves that for every nontrivial forcing $$ \mathbb{P}\in {L_\Theta }\left(\mathbb{R}\right),{1_\mathbb{P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}$$ .

This article is an introduction to combinatorics under the axiom of determinacy with a focus on partition properties and infinity Borel codes.

A set U subset of R x R is universal for countable subsets of R if and only if for all x is an element of R, the section U-x = {y is an element of R : U(x, y)} is countable and for all countable sets A subset of R, there is an x is an element of R so that U-x = A. Define the equivalence relation E-U on R by x(0) E-U x(1) if and only if U-x0 = U-x1, which is the equivalence of codes for countable sets of reals according to U. The Friedman-Stanley jump, =(+), of the equality relation takes the form E-U* where U* is the most natural Borel set that is universal for countable sets. The main result is that =(+) and E-U for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, E-U is Borel bireducible to =(+). If one assumes a particular instance of Sigma(1)(3)-generic absoluteness, then for all U subset of R x R that are Sigma(1)(1) (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of =(+) into E-U.

If X is a set, E is an equivalence relation on X, and n ∈ω, then define [X]^n_E = {(x_0, ..., x_n - 1) ∈^nX : (∀ i,j)(i ≠ j ⇒(x_i E x_j))}. For n ∈ω, a set X has the n-Jónsson property if and only if for every function f : [X]^n_= → X, there exists some Y ⊆ X with X and Y in bijection so that f[[Y]^n_=] ≠ X. A set X has the Jónsson property if and only for every function f : (⋃_n ∈ω[X]^n_=) → X, there exists some Y ⊆ X with X and Y in bijection so that f[⋃_n ∈ω [Y]^n_=] ≠ X. Let n ∈ω, X be a Polish space, and E be an equivalence relation on X. E has the n-Mycielski property if and only if for all comeager C ⊆^nX, there is some Δ_1^1 A ⊆ X so that E ≤_Δ_1^1 E ↾ A and [A]^n_E ⊆ C. The following equivalence relations will be considered: E_0 is defined on ^ω2 by x E_0 y if and only if (∃ n)(∀ k > n)(x(k) = y(k)). E_1 is defined on ^ω(^ω2) by x E_1 y if and only if (∃ n)(∀ k > n)(x(k) = y(k)). E_2 is defined on ^ω2 by x E_2 y if and only if ∑{1/n + 1 : n ∈ x y} < ∞, where denotes the symmetric difference. E_3 is defined on ^ω(^ω2) by x E_3 y if and only if (∀ n)(x(n) E_0 y(n)). Holshouser and Jackson have shown that ℝ is Jónsson under 𝖠𝖣. It will be shown that E_0 does not have the 3-Mycielski property and that E_1, E_2, and E_3 do not have the 2-Mycielski property. Under 𝖹𝖥 + 𝖠𝖣, ^ω 2 / E_0 does not have the 3-Jónsson property.

In 𝖹𝖥𝖢, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation E ∈ L(ℝ) on ℝ with all Δ_1^1 classes and every σ-ideal I on ℝ so that the associated forcing ℙ_I of I^+ Δ_1^1 subsets is proper, there exists some I^+ Δ_1^1 set C so that E ↾ C is a Δ_1^1 equivalence relation. In 𝖹𝖥 + 𝖣𝖢 + 𝖠𝖣_ℝ + V = L(𝒫(ℝ)), for every equivalence relation E on ℝ with all Δ_1^1 classes and every σ-ideal I on ℝ so that the associated forcing ℙ_I is proper, there is some I^+ Δ_1^1 set C so that E ↾ C is a Δ_1^1 equivalence relation.