Large Cardinals Research Articles

Approximation properties of torsion classes

AbstractWe strengthen a result of Bagaria and Magidor (Trans. Amer. Math. Soc. 366 (2014), no. 4, 1857–1877) about the relationship between large cardinals and torsion classes of abelian groups, and prove that the Maximum Deconstructibility principle introduced in Cox (J. Pure Appl. Algebra 226 (2022), no. 5) requires large cardinals; it sits, implication‐wise, between Vopěnka's Principle and the existence of an ‐strongly compact cardinal. While deconstructibility of a class of modules always implies the precovering property by Saorín and Šťovíček (Adv. Math. 228 (2011), no. 2, 968–1007), the concepts are (consistently) nonequivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits.

Vector multispaces and multispace codes

Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than 1 are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of Fqn) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed n and q.

Rings of formal power series and symmetric real semigroups

Sections 1–6 of this paper deal with the rings ▪ of (generalized) power series over a formally real field F of coefficients and (non-negative) exponents in an arbitrary totally ordered abelian group G. We determine explicitly the nature of the elements constituting the real spectra of these rings, its structure under the partial order of specialization, and the structure of their associated real semigroups (RS). Theorem A is a corollary of these results. Starting with section 7 we introduce a new class of symmetric real semigroups by requiring that their spaces of characters satisfy analogs to some of the basic laws holding in the real spectra of the rings ▪. These spaces have, amongst others, remarkable symmetry properties that justify their name (see 7.16 and 7.17). In §8 we prove that every finite symmetric RS is isomorphic to the RS associated to some ring of formal power series, but that such representation fails, in general, for symmetric RSs of suitably large infinite cardinality. Our study leads to showing, in §10, that the inclusion map G⁎∪{0}↪G of the reduced special group G⁎ of invertible elements of a symmetric RS G has a retract. As a consequence, every symmetric RS, G, is isomorphic to the extension of G⁎ by a 3-semigroup Δ defined in terms of that retract. Extensions of this form are introduced and studied in §9. Finally, in §11 we present a first order axiomatisation of the class of symmetric RSs in the natural language for real semigroups.

Model theoretic characterizations of large cardinals revisited

In Boney [Israel J. Math. 236 (2020), pp. 133–181], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals.

On a sumset problem for dilated integer sets

Let [Formula: see text] be a non-empty finite set of integers. For any integer [Formula: see text], let [Formula: see text]. One of the important problems in additive combinatorics is to find the optimal lower bound for the cardinality of the sumset [Formula: see text]. Cilleruelo, Silva, and Vinuesa conjectured that, if [Formula: see text] is a positive integer and [Formula: see text] is a finite set of integers with sufficiently large cardinality, then [Formula: see text]. For some values of [Formula: see text] this conjecture is already confirmed. In this article, under certain restrictions on set [Formula: see text], we confirm this conjecture for all positive integers [Formula: see text] except for those of the form [Formula: see text] or [Formula: see text] where [Formula: see text] is an odd prime number and [Formula: see text] for [Formula: see text].

Transferring compactness

AbstractWe demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a strongly inaccessible cardinal that is ‐‐stationary for all but not weakly compact. This is in sharp contrast to the situation in the constructible universe , where being ‐‐stationary is equivalent to being ‐indescribable. We also show that it is consistent that there is a cardinal such that is ‐stationary for all and , answering a question of Sakai.

On affine spaces of alternating matrices with constant rank

Let F be a field, and n ≥ r > 0 be integers, with r even. Denote by A n ⁡ ( F ) the space of all n-by-n alternating matrices with entries in F . We consider the problem of determining the greatest possible dimension for an affine subspace of A n ⁡ ( F ) in which every matrix has rank equal to r (or rank at least r). Recently Rubei [Affine subspaces of skewsymmetric matrices with constant rank. Linear Multilinear Algebra. 2023. in press. doi: 10.1080/03081087.2023.2198759.] has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that n ≥ r + 3 and | F | ≥ min ( r − 1 , r 2 + 2 ) , we also determine the affine subspaces of rank r matrices in A n ⁡ ( F ) that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case n ≤ r + 2 .

Catastrophic cascade of failures in interdependent hypergraphs.

The failures of individual agents can significantly impact the functionality of associated groups in interconnected systems. To reveal these impacts, we develop a threshold model to investigate cascading failures in double-layer hypergraphs with interlayer interdependence. We hypothesize that a hyperedge disintegrates when the proportion of failed nodes within it exceeds a threshold. Due to the interdependence between a node and its replica in the other layer, the disintegrations of these hyperedges could trigger a cascade of events, leading to an iterative collapse across these two layers. We find that double-layer hypergraphs undergo abrupt, discontinuous first-order phase transitions during systemic collapse regardless of the specific threshold value. Additionally, the connectivity measured by average cardinality and hyperdegree plays a crucial role in shaping system robustness. A higher average hyperdegree always strengthens system robustness. However, the relationship between system robustness and average cardinality exhibits non-monotonic behaviors. Specifically, both excessively small and large average cardinalities undermine system robustness. Furthermore, a higher threshold value can boost the system's robustness. In summary, our study provides valuable insights into cascading failure dynamics in double-layer hypergraphs and has practical implications for enhancing the robustness of complex interdependent systems across domains.

Clustering coefficients for networks with higher order interactions.

We introduce a clustering coefficient for nondirected and directed hypergraphs, which we call the quad clustering coefficient. We determine the average quad clustering coefficient and its distribution in real-world hypergraphs and compare its value with those of random hypergraphs drawn from the configuration model. We find that real-world hypergraphs exhibit a nonnegligible fraction of nodes with a maximal value of the quad clustering coefficient, while we do not find such nodes in random hypergraphs. Interestingly, these highly clustered nodes can have large degrees and can be incident to hyperedges of large cardinality. Moreover, highly clustered nodes are not observed in an analysis based on the pairwise clustering coefficient of the associated projected graph that has binary interactions, and hence higher order interactions are required to identify nodes with a large quad clustering coefficient.

On the consistency of ZF with an elementary embedding from Vλ+2 into Vλ+2

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal [Formula: see text] and nontrivial elementary embedding [Formula: see text]. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered. [Formula: see text] is the assertion, introduced by Hugh Woodin, that [Formula: see text] is an ordinal and there is an elementary embedding [Formula: see text] with critical point [Formula: see text]. And [Formula: see text] asserts that [Formula: see text] holds for some [Formula: see text]. The axiom [Formula: see text] is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe [Formula: see text] (in which case [Formula: see text] must be a limit ordinal), but we assume only ZF. We prove, assuming ZF [Formula: see text] [Formula: see text] [Formula: see text] “[Formula: see text] is an even ordinal”, that there is a proper class transitive inner model [Formula: see text] containing [Formula: see text] and satisfying ZF [Formula: see text] [Formula: see text] [Formula: see text] “there is an elementary embedding [Formula: see text]”; in fact we will have [Formula: see text] ⊆[Formula: see text], where [Formula: see text] witnesses [Formula: see text] in [Formula: see text]. This result was first proved by the author under the added assumption that [Formula: see text] exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also [Formula: see text] is a limit ordinal and [Formula: see text]-DC holds in [Formula: see text], then the model [Formula: see text] will also satisfy [Formula: see text]-DC. We show that ZFC [Formula: see text] “[Formula: see text] is even” [Formula: see text] [Formula: see text] implies [Formula: see text] exists for every [Formula: see text], but if consistent, this theory does not imply [Formula: see text] exists.

The second-order version of Morley’s theorem on the number of countable models does not require large cardinals

The second-order version of Morley’s theorem on the number of countable models does not require large cardinals

In search of maximum non-overlapping codes

AbstractNon-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes has been determined only for cases where the codeword length divides the size of the alphabet, and for codes with codewords of length two or three. For all other alphabet sizes and codeword lengths no computationally feasible way to identify non-overlapping codes that attain the maximum size has been found to date. Herein we characterize maximal non-overlapping codes. We formulate the maximum non-overlapping code problem as an integer optimization problem and determine necessary conditions for optimality of a non-overlapping code. Moreover, we solve several instances of the optimization problem to show that the hitherto known constructions do not generate the optimal codes for many alphabet sizes and codeword lengths. We also evaluate the number of distinct maximum non-overlapping codes.

HyBench: A New Benchmark for HTAP Databases

In this paper, we propose, HyBench, a new benchmark for HTAP databases. First, we generate the testing data by simulating a representative HTAP application. We particularly develop a time-dependent generation phase and an anomaly generation phase for testing HTAP with large cardinality and various anomalies. Second, we propose a set of hybrid workloads. Specifically, we design 18 read/write transactions, 13 analytical queries, and a mix workload of 6 analytical transactions and 6 interactive queries. We also develop a graph-based parameter curation method to control the access patterns including skew access and data contention of the hybrid workload. Third, we propose a unified metric for quantifying the overall HTAP performance. Particularly, we introduce a query-driven method that evaluates the data freshness (lag time between analytics and transactions). Then we introduce a three-phase execution rule to compute a unified metric, combining the performance of OLTP (TPS), OLAP (QPS), and OLXP (XPS) and data freshness. To verify the effectiveness of HyBench and to debunk the myth of different HTAP architectures, extensive experiments have been conducted over five HTAP databases.

Simple Primitives With Feasibility- and Contextuality-Dependence for Open-World Compositional Zero-Shot Learning.

The task of Open-World Compositional Zero-Shot Learning (OW-CZSL) is to recognize novel state-object compositions in images from all possible compositions, where the novel compositions are absent during the training stage. The performance of conventional methods degrades significantly due to the large cardinality of possible compositions. Some recent works consider simple primitives (i.e., states and objects) independent and separately predict them to reduce cardinality. However, it ignores the heavy dependence between states, objects, and compositions. In this paper, we model the dependence via feasibility and contextuality. Feasibility-dependence refers to the unequal feasibility of compositions, e.g., hairy is more feasible with cat than with building in the real world. Contextuality-dependence represents the contextual variance in images, e.g., cat shows diverse appearances when it is dry or wet. We design Semantic Attention (SA) to capture the feasibility semantics to alleviate impossible predictions, driven by the visual similarity between simple primitives. We also propose a generative Knowledge Disentanglement (KD) to disentangle images into unbiased representations, easing the contextual bias. Moreover, we complement the independent compositional probability model with the learned feasibility and contextuality compatibly. In the experiments, we demonstrate our superior or competitive performance, SA-and-kD-guided Simple Primitives (SAD-SP), on three benchmark datasets.

Weak Indestructibility and Reflection

AbstractThere is a balance between the amount of (weak) indestructibility one can have and the amount of strong cardinals. It’s consistent relative to large cardinals to have lots of strong cardinals and all of their degrees of strength are weakly indestructible. But this necessitates the destructibility of the partially strong cardinals. Guaranteeing the indestructibility of the partially strong cardinals is shown to be harder. In particular, this work establishes an equiconsistency between: 1.a proper class of cardinals that are strong reflecting strongs; and2.weak indestructibility for (κ+2)-strength for all cardinals κ in the presence of a proper class of strong cardinals.These have a much higher consistency strength than: 3.weak indestructibility for all degrees of strength for a proper class of strong cardinals.This discrepancy holds even if we weaken (2) from the presence of a proper class to just two strong cardinals. (2) is also equivalent to weak indestructibility for all λ-strength for λ far beyond (κ+2); well beyond the next measurable limit of measurables above κ, but before the next μ that is (μ+2)-strong.One direction of the equiconsistency of (1) and (2) is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed, and similar results are derived for supercompacts and supercompacts reflecting supercompacts.Abstract prepared by James Holland.E-mail: jch258@scarletmail.rutgers.eduURL: https://sites.math.rutgers.edu/~jch258/assets/dis.pdf

More Ramsey theory for highly connected monochromatic subgraphs

Abstract An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey’s theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák, and Shelah (2021, Acta Mathematica Hungarica 163, 309–322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with ZFC: • $\kappa \to _{hc} (\kappa )^2_\omega $ for every regular cardinal $\kappa \geq \aleph _2$ , • $\neg \mathsf {CH}+ \aleph _2 \to _{hc} (\aleph _1)^2_\omega $ . Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181–197), we also show that • $\aleph _2 \to _{hc} [\aleph _2]^2_{\omega ,2}$ is consistent with $\neg \mathsf {CH}$ . To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.

STATIONARY REFLECTION AND THE FAILURE OF THE SCH

AbstractIn this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu $ such that the singular cardinal hypothesis fails at $\nu $ and every collection of fewer than $\operatorname {\mathrm {cf}}(\nu )$ stationary subsets of $\nu ^{+}$ reflects simultaneously. For $\operatorname {\mathrm {cf}}(\nu )> \omega $ , this situation was not previously known to be consistent. Using different methods, we reduce the upper bound on the consistency strength of this situation for $\operatorname {\mathrm {cf}}(\nu ) = \omega $ to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.

A Cardinality Estimator in Complex Database Systems Based on TreeLSTM.

Cardinality estimation is critical for database management systems (DBMSs) to execute query optimization tasks, which can guide the query optimizer in choosing the best execution plan. However, traditional cardinality estimation methods cannot provide accurate estimates because they cannot accurately capture the correlation between multiple tables. Several recent studies have revealed that learning-based cardinality estimation methods can address the shortcomings of traditional methods and provide more accurate estimates. However, the learning-based cardinality estimation methods still have large errors when an SQL query involves multiple tables or is very complex. To address this problem, we propose a sampling-based tree long short-term memory (TreeLSTM) neural network to model queries. The proposed model addresses the weakness of traditional methods when no sampled tuples match the predicates and considers the join relationship between multiple tables and the conjunction and disjunction operations between predicates. We construct subexpressions as trees using operator types between predicates and improve the performance and accuracy of cardinality estimation by capturing the join-crossing correlations between tables and the order dependencies between predicates. In addition, we construct a new loss function to overcome the drawback that Q-error cannot distinguish between large and small cardinalities. Extensive experimental results from real-world datasets show that our proposed model improves the estimation quality and outperforms traditional cardinality estimation methods and the other compared deep learning methods in three evaluation metrics: Q-error, MAE, and SMAPE.

Large cardinals at the brink

Kunen's theorem that assuming the Axiom of Choice there are no Reinhardt cardinals is a key milestone in the program to understand large cardinal axioms. But this theorem is not the end of a story, rather it is the beginning.

ASYMMETRIC CUT AND CHOOSE GAMES

AbstractWe investigate a variety of cut and choose games, their relationship with (generic) large cardinals, and show that they can be used to characterize a number of properties of ideals and of partial orders: certain notions of distributivity, strategic closure, and precipitousness.