arXiv subjects
Publications and source records attributed to Monroe Eskew.
In his Mostowski lecture in Wroc{\l}aw in 2024, Stevo Todor\v{c}evi\'c asked whether it is consistent that Rado's Conjecture holds at two successive cardinals. We show that it is consistent that Rado's Conjecture holds at all regular cardinals.
We investigate big Ramsey degrees of finite substructures of the universal countable homogeneous meet-tree and its binary variant. We prove that structures containing antichains have infinite big Ramsey degrees, and the big Ramsey degree of a 2-element chain is at least 8 and 7 for the binary variant. We deduce that the generic C-relation does not have finite big Ramsey degrees.
We analyze some posets involved in forcing constructions for dense ideals, showing that the Anonymous Collapse and the Dual Shioya Collapse are equivalent for collapsing a large cardinal to $\omega_2$. We also give a somewhat simplified construction of a normal ideal $I$ on $\omega_2$ such that $\mathcal{P}(\omega_2)/I \sim \mathrm{Col}(\omega_1,\omega_2)$.
In this paper, we obtain the consistency, relative to large cardinals, of the existence of dense ideals on every successor of a regular cardinal simultaneously. Using a consequent transfer principle, we show that in this model there is a $\sigma$-complete, $\aleph_1$-dense ideal on $\aleph_{n+1}$ for every $n < \omega$, answering a question of Foreman. Using this construction we show the consistency of the existence of various irregular ultrafilters on $\omega_n$, the consistency of the Foreman-Laver reflection property for the chromatic number of graphs for all possible pairs of cardinals below $\aleph_\omega$, and the simultaneous consistency of the partition hypotheses $\mathrm{PH}_n(\omega_m)$ for $n < m$.
Addressing a question of Shioya, we show that two-step iterations of the Laver collapse can force saturated ideals and Chang conjectures.
Using Easton collapses, we give a simplified construction of a model in which Chang's Conjecture for triples holds.
From large cardinals we show the consistency of normal, fine, $κ$-complete $λ$-dense ideals on $\mathcal{P}_κ(λ)$ for successor $κ$. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.
We explore the possibilities for elementary embeddings $j : M \to N$, where $M$ and $N$ are models of ZFC with the same ordinals, $M \subseteq N$, and $N$ has access to large pieces of $j$. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line $\mathbb R$.
We show that the weakest versions of Foreman's minimal generic hugeness axioms cannot hold simultaneously on adjacent cardinals. Moreover, conventional forcing techniques cannot produce a model of one of these axioms.
Towards combining "compactness" and "hugeness" properties at $ω_2$, we investigate the relevance of side-conditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at $ω_2$ using Neeman's forcing. But we find a barrier to the applicability of these methods to our problem and give a counterexample to a claim of Neeman about the effects of iterating such forcing.
We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.
We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The average values so determined lie in a proper extension of the range of the original functions. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral an upper and lower standard part. These numbers can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations valid for real-valued measures. We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals.
For $μ, κ$ infinite, say $\mathcal{A}\subseteq [κ]^κ$ is a $(μ,κ)$-maximal independent family if whenever $\mathcal{A}_0$ and $\mathcal{A}_1$ are pairwise disjoint non-empty in $[\mathcal{A}]^{<μ}$ then $\bigcap\mathcal{A}_0\backslash\bigcup\mathcal{A}_1 \not= \emptyset$, $\mathcal{A}$ is maximal under inclusion among families with this property, and moreover all such Booelan combinations have size $κ$. We denote by $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ the set of all cardinalities of such families, and if non-empty, we let $\mathfrak{i}_μ(κ)$ be its minimal element. Thus, $\mathfrak{i}_μ(κ)$ (if defined) is a natural higher analogue of the independence number on $ω$ for the higher Baire spaces. In this paper, we study $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ for $μ,κ$ uncountable. Among others, we show that: (1) The property $\mathfrak{sp}_{\mathfrak i}(μ,κ)\neq\emptyset$ cannot be decided on the basis of ZFC plus large cardinals. (2) Relative to a measurable, it is consistent that: (a) $(\exists κ{>}ω) \, \mathfrak{i}_κ(κ)<2^κ$; (b) $(\exists κ{>}ω)\,κ^+<\mathfrak{i}_{ω_1}(κ)<2^κ$. To the best knowledge of the authors, this is the first example of a $(μ,κ)$-maximal independent family of size strictly between $κ^+$ and $2^κ$, for uncountable $κ$. (3) $\mathfrak{sp}_{\mathfrak i}(μ,κ)$ cannot be quite arbitrary.
We investigate the possibilities of global versions of Chang's Conjecture that involve singular cardinals. We show some $\mathrm{ZFC}$ limitations on such principles, and prove relative to large cardinals that Chang's Conjecture can consistently hold between all pairs of limit cardinals below $\aleph_{ω^ω}$.
If $κ$ is regular and $2^{<κ}\leqκ^+$, then the existence of a weakly presaturated ideal on $κ^+$ implies $\square^*_κ$. This partially answers a question of Foreman and Magidor about the approachability ideal on $ω_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $ω_2$ such that $\mathcal{P}(ω_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.
Generalizing Keisler's notion of regularity for ultrafilters, Taylor introduced degrees of regularity for ideals and showed that a countably complete nonregular ideal on $ω_1$ must be somewhere $ω_1$-dense. We prove a dichotomy about degrees of regularity for $κ$-complete ideals on successor cardinals $κ$ and apply this to show that Taylor's Theorem does not generalize to higher cardinals. In particular, the existence of a nonregular ideal on $ω_2$ does not imply the existence of an $ω_2$-dense ideal on $ω_2$. We obtain similar results for normal ideals on $\mathcal P_κ(λ)$.
We show that it is consistent relative to a huge cardinal that for all infinite cardinals $κ$, $\square_κ$ holds and there is a stationary $S \subseteq κ^+$ such that $\mathrm{NS}_{κ^+} \restriction S$ is $κ^{++}$-saturated.
We discuss a system of strengthenings of "$\aleph_ω$ is Jónsson" indexed by real numbers, and identify a strongest one. We give a proof of a theorem of Silver and show that there is a barrier to weakening its hypothesis.