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Stevo Todorcevic

Publications and source records attributed to Stevo Todorcevic.

At least 37 records · Page 2Linked to original sources
arXiv2021

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.

math.LO↗
arXiv2021

Cofinal types on $ω_2$

In this paper we start the analysis of the class $\mathcal D_{\aleph_2}$, the class of cofinal types of directed sets of cofinality at most $\aleph_2$. We compare elements of $\mathcal D_{\aleph_2}$ using the notion of Tukey reducibility. We isolate some simple cofinal types in $\mathcal D_{\aleph_2}$, and then proceed to show which of these types have an immediate successor in the Tukey ordering of $\mathcal D_{\aleph_2}$.

math.LO↗
arXiv2020

The Strength of Menger's Conjecture

Menger conjectured that subsets of R with the Menger property must be $σ$-compact. While this is false when there is no restriction on the subsets of R, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We note that in fact, Menger's conjecture for projective sets has consistency strength of only an inaccessible cardinal.

math.GN↗
arXiv2019

Compact Sets of Baire Class One Functions and Maximal Almost Disjoint Families

We provide a proof that analytic almost disjoint families of infinite sets of integers cannot be maximal using a result of Bourgain about compact sets of Baire class one functions. Inspired by this and related ideas, we then provide a new proof of that there are no maximal almost disjoint families in Solovay's model. We then use the ideas behind this proof to provide an extension of a dichotomy result by Rosenthal and by Bourgain, Fremlin and Talagrand to general pointwise bounded functions in Solovay's model. We then show that the same conclusions can be drawn about the model obtained when we add a generic selective ultrafilter to the Solovay model.

math.LO↗
arXiv2018

Proof of a Conjecture of Galvin

We prove that if the set of unordered pairs of real numbers is colored by finitely many colors, there is a set of reals homeomorphic to the rationals whose pairs have at most two colors. Our proof uses large cardinals and it verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.

math.LO↗
arXiv2018

The Open Graph Axiom and Menger's Conjecture

Menger conjectured that subsets of $\mathbb R$ with the Menger property must be $σ$-compact. While this is false when there is no restriction on the subsets of $\mathbb R$, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We show that the perfect set version of the Open Graph Axiom for projective sets of reals, with consistency strength only an inaccessible cardinal, also implies Menger's conjecture restricted to this family of subsets of $\mathbb R$.

math.LO↗
arXiv2017

Homogeneous families on trees and subsymmetric basic sequences

We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal $κ$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $κ$ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic structure of the spaces. The collections of families are of independent interest and their existence is proved inductively. The fundamental stepping up argument is the analysis of such collections of families on trees.

math.FA↗
arXiv2016

Trees and gaps from a construction scheme

We present natural constructions of trees and gaps using a quite general construction scheme. In particular, we solve a natural problem about $(ω_1, ω_1)$-gaps. As it is well known $(ω_1, ω_1)$-gaps can sometimes be filled in $ω_1$-preserving forcing extensions of the set-theoretic universe. There are two natural conditions, dubbed $S$ and $T$ below, that guarantee the existence of such forcing extensions. The condition $T$ is a natural strengthening of the condition $S$ and was motivated by the numerous analogies between $(ω_1,ω_1)$-gaps and certain trees of height $ω_1.$ It turns out that the condition $S$ is in fact equivalent to the existence of such forcing extensions but we show that the condition $T$ is strictly stronger by proving that it is consistent that there are fillable $(ω_1, ω_1)$-gaps (i.e., S-gaps) but no T-gaps.

math.LO↗
arXiv2016

Rosenthal compacta that are premetric of finite degree

We show that if a separable Rosenthal compactum $K$ is an $n$-to-one preimage of a metric compactum, but it is not an $n-1$-to-one preimage, then $K$ contains a closed subset homeomorphic to either the $n-$Split interval $S_n(I)$ or the Alexandroff $n-$plicate $D_n(2^\mathbb{N})$. This generalizes a result of the third author that corresponds to the case $n=2$.

math.GN↗
arXiv2016

Suslin trees, the bounding number, and partition relations

We investigate the unbalanced ordinary partition relations of the form $λ\rightarrow {(λ, α)}^{2}$ for various values of the cardinal $λ$ and the ordinal $α$. For example, we show that for every infinite cardinal $κ,$ the existence of a $κ^{+}-$Suslin tree implies $κ^{+} \not\rightarrow {\left( κ^{+}, {\log}_κ(κ^{+}) + 2 \right)}^{2}$. The consistency of the positive partition relation $\mathfrak{b} \rightarrow {(\mathfrak{b}, α)}^{2}$ for all $α< ω_{1}$ for the bounding number $\mathfrak{b}$ is also established from large cardinals.

math.LO↗
arXiv2015

Forcing with matrices of countable elementary submodels

We analyze the forcing notion $\mathcal P$ of finite matrices whose rows consists of isomorphic countable elementary submodels of a given structure of the form $H_θ$. We show that forcing with this poset adds a Kurepa tree $T$. Moreover, if $\mathcal P_c$ is a suborder of $\mathcal P$ containing only continuous matrices, then the Kurepa tree $T$ is almost Souslin, i.e. the level set of any antichain in $T$ is not stationary in $ω_1$.

math.LO↗
arXiv2014

A disjoint union theorem for trees

We prove an infinitary disjoint union theorem for level products of trees. To implement the proof we develop a Hales-Jewett type result for words indexed by a level product of trees.

math.CO↗
arXiv2013

Combinatorial dichotomies and cardinal invariants

Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant $\mathfrak{x}$ such that the statement that $\mathfrak{x} > ω_{1}$ is equivalent to the statement that 1, $ω$, $ω_{1}$, $ω\times ω_{1}$, and ${\left[ω_{1}\right]}^{< ω}$ are the only cofinal types of directed sets of size at most ${\aleph}_{1}$. We investigate the corresponding problem for the partition relation $ω_{1} \rightarrow (ω_{1}, α)^2$ for all $α< ω_{1}$. To this effect, we investigate partition relations for pairs of comparable elements of a coherent Suslin tree $\mathbb{S}$. We show that a positive partition relation for such pairs follows from the maximal amount of the proper forcing axiom compatible with the existence of $\mathbb{S}$. As a consequence we conclude that after forcing with the coherent Suslin tree $\mathbb{S}$ over a ground model satisfying this relativization of the proper forcing axiom, $ω_{1} ~\rightarrow {(ω_{1}, α)}^{2}$ for all $α< ω_{1}$. We prove that this positive partition relation for $\mathbb{S}$ cannot be improved by showing in $\mathrm{ZFC}$ that $\mathbb{S} \not\rightarrow ({\aleph}_{1}, ω+2)^2$.

math.LO↗
arXiv2012

Finite basis for analytic strong n-gaps

We identify the finite list of minimal analytic n-gaps which are not countably separated, and we prove that every analytic n-gap which is not countably separated contains a gap from our finite list.

math.LO↗
arXiv2012

Subsets of Products of Finite Sets of Positive Upper Density

In this note we prove that for every sequence $(m_q)_{q}$ of positive integers and for every real $0<δ\leqslant1$ there is a sequence $(n_q)_{q}$ of positive integers such that for every sequence $(H_q)_{q}$ of finite sets such that $|H_q|=n_q$ for every $q\in\mathbb{N}$ and for every $D\subseteq \bigcup_k\prod_{q=0}^{k-1}H_q$ with the property that $$\limsup_k \frac{|D\cap \prod_{q=0}^{k-1} H_q|}{|\prod_{q=0}^{k-1}H_q|}\geqslantδ$$ there is a sequence $(J_q)_{q}$, where $J_q\subseteq H_q$ and $|J_q|=m_q$ for all $q$, such that $\prod_{q=0}^{k-1}J_q\subseteq D$ for infinitely many $k.$ This gives us a density version of a well-known Ramsey-theoretic result. We also give some estimates on the sequence $(n_q)_{q}$ in terms of the sequence of $(m_q)_{q}$.

math.CO↗