arXiv subjects
Publications and source records attributed to Stevo Todorčević.
We give a general construction of topological groups from combinatorial structures such as trees, towers, gaps, and subadditive functions. We connect topological properties of corresponding groups with combinatorial properties of these objects. For example, the group built from an $ω_1$-tree is Frechét iff the tree is Aronszajn. We also determine cofinal types of some of these groups under certain set theoretic assumptions.
The poset of copies of a relational structure ${\mathbb X}$ is the partial order ${\mathbb P} ({\mathbb X} ) := \langle \{ Y \subset X: {\mathbb Y} \cong {\mathbb X}\}, \subset \rangle$ and each similarity of such posets (e.g. isomorphism, forcing equivalence) determines a classification of structures. We consider the countable ultrahomogeneous tournaments: ${\mathbb Q} $ (the rational line), ${\mathbb S} (2)$ (the circular tournament), and ${\mathbb T} ^\infty$ (the random tournament); as well as the ultrahomogeneous digraphs ${\mathbb S} (3)$, ${\mathbb Q} [{\mathbb I}_n]$, ${\mathbb S} (2)[{\mathbb I}_n]$ and ${\mathbb T} ^\infty [{\mathbb I}_n]$ from Cherlin's list. If ${\mathbb G} _{\mathrm{Rado}}$ (resp. ${\mathbb Q} _n$) denotes the countable homogeneous universal graph (resp. $n$-labeled linear order), it turns out that ${\mathbb P} ({\mathbb T} ^\infty)\cong {\mathbb P} ({\mathbb G}_{\mathrm{Rado}})$ and that ${\mathbb P} ({\mathbb Q} _n)$ densely embeds in ${\mathbb P} ({\mathbb S} (n))$, for $n\in\{ 2,3\}$. Consequently, ${\mathbb B} _{\mathbb X} \cong {\mathrm{ro}}\, ({\mathbb S} \ast π)$, where ${\mathbb S}$ is the Sacks forcing and $1_{\mathbb S} \Vdash "π$ is a separative, atomless and $σ$-closed forcing", whenever ${\mathbb X}$ is a countable structure equimorphic with ${\mathbb Q}$, ${\mathbb Q} _n$, ${\mathbb S} (2)$, ${\mathbb S} (3)$, ${\mathbb Q} [{\mathbb I}_n]$ or ${\mathbb S} (2)[{\mathbb I}_n]$. Also, ${\mathbb B} _{\mathbb X} \cong {\mathrm{ro}}\, ({\mathbb S} \ast π)$, where $1_{\mathbb S} \Vdash "π$ is an $ω$-distributive forcing", whenever ${\mathbb X}$ is a countable graph embedding ${\mathbb G} _{\mathrm{Rado}}$, or a countable tournament embedding ${\mathbb T} ^\infty$, or ${\mathbb X} ={\mathbb T} ^\infty [{\mathbb I}_n]$.
We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality aleph-one the property is equivalent to the fact that the space of ultrafilters is not Corson compact. We also consider the pointwise topology on a Boolean algebra, proving a result on the Lindelöf number in the context of the ultrafilter selection property. Finally, we discuss poset Boolean algebras, interval algebras, and semilattices in the context of ultrafilter selection properties.
We construct an Asplund Banach space $\mathcal{X}$ with a norming Markuševič basis such that $\mathcal{X}$ is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact space, that also solves a question due to Wiesław Kubiś and Arkady Leiderman.
We show that there is no simple (e.g. finite or countable) basis for Borel graphs with infinite Borel chromatic number. In fact, it is proved that the closed subgraphs of the shift graph on $[\mathbb{N}]^{<\mathbb{N}}$ with finite (or, equivalently, $\leq 3$) Borel chromatic number form a $\mathbfΣ^1_2$-complete set. This answers a question of Kechris and Marks and strengthens several earlier results.
We show that the forcing axiom for countably compact, $ω_2$-Knaster, well-met posets is inconsistent. This is supplemental to an inconsistency result of Shelah and sets a new limit to the generalization of Martin's Axiom to the stage of $ω_2$.
We show that all maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if $\mathsf{MA}_\mathfrak c (\mathcal P(ω)/\mathrm{fin})$ holds. We further study the question whether there is a maximal almost disjoint family whose hyperspace is pseudocompact and prove that consistently such families do not exist \emph{genericaly}, by constructing a consistent example of a maximal almost disjoint family $\mathcal A$ of size less than $\mathfrak c$ whose hyperspace is not pseudocompact.
Let $G$ be a countable graph containing a copy of the countable random graph (Erdős-Rényi graph, Rado graph), $Emb (G)$ the monoid of its self-embeddings, ${\mathbb P} (G)=\{f[G]: f\in Emb (G)\}$ the set of copies of $G$ contained in $G$, and ${\mathcal I}_G$ the ideal of subsets of $G$ which do not contain a copy of $G$. We show that the poset $< {\mathbb P} (G), \subset>$, the algebra $P (G)/{\mathcal I}_G$, and the inverse of the right Green's pre-order $< Emb (G),\preceq ^R >$ have the 2-localization property. The Boolean completions of these pre-orders are isomorphic and satisfy the following law: for each double sequence $[b_{nm}: < n, m > \in ω\times ω]$ of elements of ${\mathbb B}$ $$\textstyle \bigwedge_{n \in ω}\; \bigvee_{m \in ω}\; b_{nm} = \bigvee_{{\mathcal T} \,\in \, Bt ({}^{<ω}ω)}\; \bigwedge_{n \in ω}\; \bigvee_{φ\,\in \,{\mathcal T} \cap {}^{n+1}ω}\; \bigwedge_{k\leq n}\; b_{kφ(k)}, $$ where $Bt ({}^{<ω}ω)$ denotes the set of all binary subtrees of the tree ${}^{<ω}ω$.
Let $(R, \sim )$ be the Rado graph, $Emb (R)$ the monoid of its self-embeddings, $Π(R)=\{ f[R]: f\in Emb (R)\}$ the set of copies of $R$ contained in $R$, and ${\mathcal I}_R$ the ideal of subsets of $R$ which do not contain a copy of $R$. We consider the poset $( Π(R ), \subset )$, the algebra $P (R)/{\mathcal I _R}$, and the inverse of the right Green's pre-order on $Emb (R)$, and show that these pre-orders are forcing equivalent to a two step iteration of the form $P \ast π$, where the poset $P$ is similar to the Sacks perfect set forcing: adds a generic real, has the $\aleph _0$-covering property and, hence, preserves $ω_1$, has the Sacks property and does not produce splitting reals, while $π$ codes an $ω$-distributive forcing. Consequently, the Boolean completions of these four posets are isomorphic and the same holds for each countable graph containing a copy of the Rado graph.