Oscillating subalgebras of the atomless countable Boolean algebra
We show that the big Ramsey degree of the Boolean algebra with 3 atoms within the countable atomless Boolean algebra is infinite.
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arXiv subjects
Stevo Todorcevic
Publications and source records attributed to Stevo Todorcevic.
Strong colorings based on oscillations
We show that for any uncountable cardinal $κ$, there is a coloring $c: [κ]^2\to ω$ such that $c''A \otimes B = ω$ for any $A, B\subseteq κ$ of order type $ω_1$ that are stationary in their common supremum. In particular, the stationary version of Erdős-Rado theorem and the higher dimensional Friedman's property are both inconsistent. We demonstrate that the theorem is optimal in various ways.
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Square-bracket operations clubs
This paper continues the investigation of the three square-bracket operations $[\cdot\cdot]$ from chapter 5 of \cite{Walks}. \ We say that a square-bracket operation $[\cdot\cdot]$ has the \emph{Ramsey club property} if for every club $C\subseteqω_{1}$, there is an uncountable subset $W$ $\subseteq ω_{1}$ such that $\left[ αβ\right] \in C$ for every $α,β\in W.$ \ The second author proved that the Proper Forcing Axiom\textsf{ }implies that all the square-bracket operations induced by Aronszajn trees have this property. We extend this result to the other two classes. We conclude that each of the statements \textquotedblleft all square-bracket operations have the Ramsey club property\textquotedblright\ and \textquotedblleft No square-bracket operation has the Ramsey club property\textquotedblright\ are consistent with \textsf{ZFC. }In other words, \textsf{ZFC }is unable to decide the status of the Ramsey club property for any square-bracket operation. Furthermore, we analyze the status of the Ramsey club property for square-bracket operations under Martin's Axiom and the Continuum Hypothesis.
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Forcing Axioms and construction schemes
We continue the development of the theory of construction schemes over $ω_1$ as introduced by the third author by studying their relation with forcing axioms. Formally, we introduce the cardinals $\mathfrak{m}^n_{\mathcal{F}}$ and use the consistency of $\mathfrak{m}^2_\mathcal{F}>ω_1$ to prove a fundamental result relating gaps and almost disjoint families over $ω$. The cardinals $\mathfrak{m}_\mathcal{F}$ are also used to prove some limiting results for contstruction schemes, some of which answer questions from \cite{schemescruz}. Finally, we show that PID implies the non-existence of $2$-capturing schemes.
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On Big Ramsey degrees of universal $ω$-edge-labeled hypergraphs
We show that the big Ramsey degrees of every countable universal $u$-uniform $ω$-edge-labeled hypergraph are infinite for every $u\geq 2$. Together with a recent result of Braunfeld, Chodounský, de Rancourt, Hubička, Kawach, and Konečný this finishes full characterisation of unrestricted relational structures with finite big Ramsey degrees.
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Basis problem for analytic multiple gaps
A k-gap is a finite k-sequence of pairwise disjoint monotone families of infinite subsets of N mixed in such a way that we cannot find a partition of N such that each family is trival on one piece of the partition. We prove that, relative to the comparison given by restriction to infinite subsets of N, for every positive integer k there is a finite basis for the class of all analytic k-gaps . We also build the fine structure theory of analytic k-gaps and give some applications. The content of Chapter 1 of this manuscript have been published as: A. Avilés, S. Todorcevic, Finite basis for analytic multiple gaps, Publ. Math. IHES. 121 (2015), 57-79. The content of Chapter 2 (except some technical results from 2.5 and 2.6) and Section 3.1, largely revised and improved, has ben published as: A. Avilés, S. Todorcevic, Types in the n-adic tree and minimal analytic gaps, Adv. Math. 292 (2016), 558-600. The content of Sections 3.4, 4.1 and 4.3 have been published as: A. Avilés, S. Todorcevic, Isolating subgaps of a multiple gap, Monatsh. Math. 186 (2018), 373--392. The rest of contents may appear elsewhere.
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Higher Dimensional Chain Conditions
We investigate higher dimensional chain conditions, where the largeness notion is given by Fubini products of a given ideal. From strong saturation properties of an ideal, we derive abstractly versions of higher dimensional $Δ$-system lemma, which imply many posets, including any finite support iteration of $σ$-centered posets and measure algebras, satisfy the higher dimensional chain conditions. We then show that if a poset satisfies a strengthening of the $σ$-finite chain condition by Horn and Tarski, then it satisfies higher dimensional chain conditions. As an application, we derive Ramsey-theoretic consequences, namely various partition hypotheses as studied by Bannister, Bergfalk, Moore and Todorcevic, from the existence of ideals satisfying strong chain conditions.
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Dense metrizable subspaces in powers of Corson compacta
We characterize when the countable power of a Corson compactum has a dense metrizable subspace and construct consistent examples of Corson compacta whose countable power does not have a dense metrizable subspace. We also give several remarks about ccc Corson compacta and, as a byproduct, we obtain a new proof of Kunen and van Mill's characterization of when a Corson compactum supporting a strictly positive measure is metrizable.
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Infinite dimensional sequential compactness: Sequential compactness based on barriers
We introduce a generalization of sequential compactness using barriers on $ω$ extending naturally the notion introduced in [W. Kubiś and P. Szeptycki, On a topological Ramsey theorem, \emph{Canad. Math. Bull.}, 66 (2023), {156}--{165}]. We improve results from [C. Corral and O. Guzm{á}n and C. L{ó}pez-Callejas, High dimensional sequential compactness, \emph{Fund. Math.}] by building spaces that are $\mathcal{B}$-sequentially compact but no $\mathcal{C}$-sequentially compact when the barriers $\mathcal{B}$ and $\mathcal{C}$ satisfy certain rank assumption which turns out to be equivalent to a Katětov-order assumption. Such examples are constructed under the assumption $\mathfrak{b} =\mathfrak{c}$. We also exhibit some classes of spaces that are $\mathcal{B}$-sequentially compact for every barrier $\mathcal{B}$, including some classical classes of compact spaces from functional analysis, and as a byproduct we obtain some results on angelic spaces. Finally we introduce and compute some cardinal invariants naturally associated to barriers.
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Aronszajn Free Kurepa Trees
We consider a transitive relation on the power set of $ω_1$ and show if there is a maximal element with respect to this relation then there is a Kurepa tree with no Aronszajn subtree. We also show that if there is a maximal subset of $ω_1$, then there are Kurepa trees which are not club isomorphic. These maximal subsets of $ω_1$ exist in many known models that are obtained from the constructible universe without large cardinal assumptions. For instance, whenever $α_0 \in ω_1$ and $X \subset ω_1$ are such that $ω_1^{\textsc{L}[X \cap α_0]} = ω_1, ω_2^{\textsc{L}[X]} = ω_2$ and $\textsc{V}$ is a semiproper forcing extension of $\textsc{L}[X]$ then $X$ is maximal in $\textsc{V}$.
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Can You Take Komjath's Inaccessible Away?
In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains an Aronszajn subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem that asserts the same consistency from two inaccessible cardinals. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree $T$ such that if $U \subset T$ is a Kurepa tree with the inherited order from $T$, then $U$ has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: assume $\textrm{MA}_{ω_2}$ holds and $ω_2$ is not a Mahlo cardinal in $\textsc{L}$. Then there is a Kurepa tree with the property that every Kurepa subset has an Aronszajn subtree. Our work entails proving a new lemma about Todorcevic's $ρ$ function which might be useful in other contexts.
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A descriptive approach to higher derived limits
We present a new aspect of the study of higher derived limits. More precisely, we introduce a complexity measure for the elements of higher derived limits over the directed set $Ω$ of functions from $\mathbb{N}$ to $\mathbb{N}$ and prove that cocycles of this complexity are images of cochains of the roughly the same complexity. In the course of this work, we isolate a partition principle for powers of directed sets and show that whenever this principle holds, the corresponding derived limit $\mathrm{lim}^n$ is additive; vanishing results for this limit are the typical corollary. The formulation of this partition hypothesis synthesizes and clarifies several recent advances in this area.
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Construction schemes: transferring structures from $ω$ to $ω_1$
A structural analysis of construction schemes is developed. That analysis is used to give simple and new constructions of combinatorial objects which have been of interest to set theorists and topologists. We then continue the study of capturing axioms associated to construction schemes. From them, we deduce the existence of several uncountable structures which are known to be independent from the usual axioms of Set Theory. Lastly, we prove that the capturing axiom $FCA(part)$ is implied by Jensen's $\Diamond$ principle.
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A new small Dowker space
It is proved that if there exists a Luzin set, or if either the stick principle or diamond(b) hold, then a strong instance of the guessing principle $\clubsuit_{AD}$ holds at the first uncountable cardinal. In particular, any of the above hypotheses entails the existence of a Dowker space of size $\aleph_1$.
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A Dual Ramsey theorem for trees
We prove a dualization of the Graham--Rothschild Theorem for variable words indexed by homogeneous trees.
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A combinatorial property of rho-functions
We show that if $\mathcal{T}$ is any Hausdorff topology on $ω_{1}$, then any subset of $ω_{1}$ which is homeomorphic to the rationals under $\mathcal{T}$ can be refined to a homeomorphic copy of the rationals on which $\barρ$ is shift-increasing.
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Galvin's problem in higher dimensions
It is proved that for each natural number $n$, if $\left| \mathbb{R} \right| = {\aleph}_{n}$, then there is a coloring of ${\left[ \mathbb{R} \right]}^{n+2}$ into ${\aleph}_{0}$ colors that takes all colors on ${\left[ X \right]}^{n+2}$ whenever $X$ is any set of reals which is homeomorphic to $\mathbb{Q}$. This generalizes a theorem of Baumgartner and sheds further light on a problem of Galvin from the 1970s. Our result also complements and contrasts with our earlier result saying that any coloring of ${\left[ \mathbb{R} \right]}^{2}$ into finitely many colors can be reduced to at most $2$ colors on the pairs of some set of reals which is homeomorphic to $\mathbb{Q}$ when large cardinals exist.
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Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals
We define a collection of topological Ramsey spaces consisting of equivalence relations on $ω$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $ω$. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.
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