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Assaf Rinot

Publications and source records attributed to Assaf Rinot.

At least 19 recordsLinked to original sources
arXiv2026

More notions of forcing add a square

Foreman and Magidor showed that the continuum hypothesis implies the existence of a countably-closed $\aleph_2$-cc forcing notion $\mathbb P$ for adding $\square_{\aleph_1}$. Here, we show that $\mathbb P$ may consistently be realized as an $\aleph_2$-Souslin tree. More generally, we prove that $\square_\lambda$ may be added by a $\lambda^+$-Souslin tree, providing the first analog of the Foreman--Magidor forcing at the level of successors of singular cardinals. Our construction is uniform and extends to inaccessible cardinals as well.

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arXiv2025

The power of trees

We give two consistent constructions of trees $T$ whose finite power $T^{n+1}$ is sharply different from $T^n$: 1. An $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact. 2. For an inaccessible $\kappa$ and a positive integer $n$, a $\kappa$-tree such that all of its $n$-derived trees are Souslin and all of its $(n+1)$-derived trees are special.

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arXiv2025

A new model for all $C$-sequences are trivial

We construct a model in which all $C$-sequences are trivial, yet there exists a $\kappa$-Souslin tree with full vanishing levels. This answers a question of Lambie-Hanson and Rinot, and provides an optimal combination of compactness and incompactness. It is obtained by incorporating a so-called mutually exclusive ascent path to Kunen's original forcing construction.

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arXiv2024

A model for global compactness

In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which $\aleph_{\omega+1}$ carries a uniform ultrafilter that is $\theta$-indecomposable for every uncountable cardinal $\theta<\aleph_\omega$. In this paper, we give a global version of this result, as follows: Assuming the consistency of a supercompact cardinal, we produce a model of set theory in which for every singular cardinal $\lambda$, there exists a uniform ultrafilter on $\lambda^+$ that is $\theta$-indecomposable for every cardinal $\theta$ such that $cf(\lambda)<\theta<\lambda$. In our model, many instances of compactness for chromatic numbers hold, from which we infer that Hajnal's gap-1 counterexample to Hedetniemi's conjecture is best possible on the grounds of ZFC.

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arXiv2024

Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines

It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with his Basis Theorem for linear orders of size $\aleph_1$. The proof combines walks on ordinals, club guessing, strong colourings of three different types, and a bit of finite combinatorics. This and further non-structure theorems for Aronszajn lines and trees are established for successors of regulars, successors of singulars, as well as inaccessibles.

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arXiv2024

Ketonen's question and other cardinal sins

Answering a question of Ketonen from the late 1970's, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing the limit of a decreasing sequence of models of ZFC. The utility of this proof technique is demonstrated further in this paper, where a problem by Bagaria and Magidor concerning strong compactness, and a problem by Lambie-Hanson and Rinot concerning the $C$-sequence number are solved as well.

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arXiv2024

Diamond on Kurepa trees

We introduce a new weak variation of diamond that is meant to only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by proving it is compatible with Martin's axiom. We then prove that this principle is nontrivial by showing it may consistently fail.

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arXiv2024

Proxy principles in combinatorial set theory

The parameterized proxy principles were introduced by Brodsky and Rinot in a 2017 paper, as new foundations for the construction of $\kappa$-Souslin trees in a uniform way that does not depend on the nature of the (regular uncountable) cardinal $\kappa$. Since their introduction, these principles have facilitated construction of Souslin trees with complex combinations of features, and have enabled the discovery of completely new scenarios in which Souslin trees must exist. Furthermore, the proxy principles have found new applications beyond the construction of trees. This paper opens with a comprehensive exposition of the proxy principles. We motivate their very definition, emphasizing the utility of each of the parameters and the consequent flexibility that they provide. We then survey the findings surrounding them, presenting a rich spectrum of unrelated models and configurations in which the proxy principles are known to hold, and showcasing a gallery of Souslin trees constructed from the principles. The last two sections of the paper offer new results. In particular, for every positive integer $n$, we give a construction of a $\lambda^+$-Souslin tree all of whose $n$-derived trees are Souslin, but whose $(n+1)$-power is special.

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arXiv2024

Full Souslin trees at small cardinals

A $κ$-tree is said to be full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $κ$-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be $\aleph_3$ many full $\aleph_2$-trees such that the product of any countably many of them is an $\aleph_2$-Souslin tree.

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arXiv2024

Squares, ultrafilters and forcing axioms

We study relationships between various set theoretic compactness principles, focusing on the interplay between the three families of combinatorial objects or principles mentioned in the title. Specifically, we show the following. (1) Strong forcing axioms, in general incompatible with the existence of indexed squares, can be made compatible with weaker versions of indexed squares. (2) Indexed squares and indecomposable ultrafilters with suitable parameters can coexist. As a consequence, the amount of stationary reflection known to be implied by the existence of a uniform indecomposable ultrafilter is optimal. (3) The Proper Forcing Axiom implies that any cardinal carrying a uniform indecomposable ultrafilter is either measurable or a supremum of countably many measurable cardinals. Leveraging insights from the preceding sections, we demonstrate that the conclusion cannot be improved.

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arXiv2024

Diamond on ladder systems and countably metacompact topological spaces

The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $Δ$-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $L$ over the first uncountable cardinal for which the corresponding space $X_L$ is not a $Δ$-space, and asked whether there is a ZFC example of a ladder system $L$ over some cardinal $κ$ for which $X_L$ is not countably metacompact, in particular, not a $Δ$-space. We prove that an affirmative answer holds for the cardinal $κ=cf(\beth_{ω+1})$. Assuming $\beth_ω=\aleph_ω$, we get an example at a much lower cardinal, namely $κ=2^{2^{2^{\aleph_0}}}$, and our ladder system $L$ is moreover $ω$-bounded.

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arXiv2023

Was Ulam right? II: Small width and general ideals

We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam's theorem and its extension by Hajnal, it is proved that if $κ$ is a regular uncountable cardinal that is not weakly compact in L, then there is a universal witness for non-weak-saturation of $κ$-complete ideals. Specifically, there are $κ$-many decompositions of $κ$ such that, for every $κ$-complete ideal $J$ over $κ$, and every $B\in J^+$, one of the decompositions shatters $B$ into $κ$-many $J^+$-sets. A second focus here is the feature of narrowness of colourings, one already present in the theorem of Sierpinski. This feature ensures that a colouring suitable for an ideal is also suitable for all superideals possessing the requisite completeness degree. It is proved that unlike successors of regulars, every successor of a singular cardinal admits such a narrow colouring.

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arXiv2023

The vanishing levels of a tree

We initiate the study of the spectrum $Vspec(κ)$ of sets that can be realized as the vanishing levels $V(T)$ of a normal $κ$-tree $T$. The latter is an invariant in the sense that if $T$ and $T'$ are club-isomorphic, then the symmetric difference of $V(T)$ and $V(T')$ is nonstationary. Additional features of this invariant imply that $Vspec(κ)$ is closed under finite unions and intersections. The set $V(T)$ must be stationary for an homogeneous normal $κ$-Aronszajn tree $T$, and if there exists a special $κ$-Aronszajn tree, then there exists one $T$ that is homogeneous and satisfies $V(T)=κ$ (modulo clubs). It is consistent (from large cardinals) that there is an $\aleph_2$-Souslin tree, and yet $V(T)$ is co-stationary for every $\aleph_2$-tree $\mathbf T$. Both $V(T)=\emptyset$ and $V(T)=κ$ (modulo clubs) are shown to be feasible using $κ$-Souslin trees even at some large cardinal close to a weakly compact. It is also possible to have a family of $2^κ$ many $κ$-Souslin trees for which the corresponding family of vanishing levels forms an antichain modulo clubs.

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arXiv2023

A counterexample related to a theorem of Komjath and Weiss

In a paper from 1987, Komjath and Weiss proved that for every regular topological space $X$ of character less than $\mathfrak b$, if $X\rightarrow(top~ω+1)^1_ω$, then $X\rightarrow(top~α)^1_ω$ for all $α<ω_1$. In addition, assuming $\diamondsuit$, they constructed a space $X$ of size continuum, of character $\mathfrak b$, satisfying $X\rightarrow(top~ω+1)^1_ω$, but not $X\rightarrow(top~ω^2+1)^1_ω$. Here, a counterexample space with the same characteristics is obtained outright in ZFC.

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arXiv2023

A Shelah group in ZFC

In a paper from 1980, Shelah constructed an uncountable group all of whose proper subgroups are countable. Assuming the continuum hypothesis, he constructed an uncountable group $G$ that moreover admits an integer $n$ satisfying that for every uncountable $X\subseteq G$, every element of $G$ may be written as a group word of length $n$ in the elements of $X$. The former is called a Jonsson group and the latter is called a Shelah group. In this paper, we construct a Shelah group on the grounds of ZFC alone, that is, without assuming the continuum hypothesis. More generally, we identify a combinatorial condition (coming from the theories of negative square-bracket partition relations and strongly unbounded subadditive maps) sufficient for the construction of a Shelah group of size $κ$, and prove that the condition holds true for all successors of regular cardinals (such as $κ=\aleph_1,\aleph_2,\aleph_3,\ldots$). This also yields the first consistent example of a Shelah group of size a limit cardinal.

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arXiv2023

Sums of triples in Abelian groups

Motivated by a problem in additive Ramsey theory, we extend Todorcevic's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for every Abelian group $G$ of size $\aleph_2$, there exists a coloring $c:G\rightarrow\mathbb Z$ such that for every uncountable $X\subseteq G$ and every integer $k$, there are three distinct elements $x,y,z$ of $X$ such that $c(x+y+z)=k$.

math.LO↗
arXiv2022

Sigma-Prikry forcing III: Down to Aleph_omega

We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_ω$ together with the reflection of all stationary subsets of $\aleph_{ω+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.

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