Dániel T. Soukup
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View article: Towers and gaps at uncountable cardinals
Towers and gaps at uncountable cardinals Open
Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $…
View article: MORE ZFC INEQUALITIES BETWEEN CARDINAL INVARIANTS
MORE ZFC INEQUALITIES BETWEEN CARDINAL INVARIANTS Open
Motivated by recent results and questions of Raghavan and Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show tha…
View article: Representative & Fair Synthetic Data
Representative & Fair Synthetic Data Open
Algorithms learn rules and associations based on the training data that they are exposed to. Yet, the very same data that teaches machines to understand and predict the world, contains societal and historic biases, resulting in biased algo…
View article: Idea univerzity z české perspektivy
Idea univerzity z české perspektivy Open
The aim of this book is to explore the ‘idea of a university’ and specific modern developments both abroad and in the Czech Republic through dialogues with prominent Czech academics: Stanislav Balík, Petr Dvořák, Petr Fiala, Pavel Floss, J…
View article: On the complexity of classes of uncountable structures: trees on $\aleph _1$
On the complexity of classes of uncountable structures: trees on $\aleph _1$ Open
We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire-space $ω_1^{ω_1}$. First, we will show that none of these classes have the Baire property (unless they a…
View article: Extremal triangle-free and odd-cycle-free colourings of uncountable\n graphs
Extremal triangle-free and odd-cycle-free colourings of uncountable\n graphs Open
The optimality of the Erd\\H{o}s-Rado theorem for pairs is witnessed by the\ncolouring $\\Delta_\\kappa : [2^\\kappa]^2 \\rightarrow \\kappa$ recording the least\npoint of disagreement between two functions. This colouring has no\nmonochro…
View article: The open dihypergraph dichotomy and the second level of the Borel hierarchy
The open dihypergraph dichotomy and the second level of the Borel hierarchy Open
We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the $\aleph_0$-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the perf…
View article: Reducing the dichromatic number via cycle reversions in infinite\n digraphs
Reducing the dichromatic number via cycle reversions in infinite\n digraphs Open
We prove the following conjecture of S. Thomass\\'e: for every (potentially\ninfinite) digraph $ D $ it is possible to iteratively reverse directed cycles\nin such a way that the dichromatic number of the final reorientation $ D^{*} $\nof …
View article: A 0-dimensional, Lindel\"of space that is not strongly D
A 0-dimensional, Lindel\"of space that is not strongly D Open
A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\{U_x:x\in X\}$, there is a $D\subseteq X$ such that $\{U_x:x\in D\}$ covers $X$ and $D$ is locally finite in the topology generated by $\{U_x:x\in X\}$. We prove…
View article: A 0-dimensional, Lindelöf space that is not strongly D
A 0-dimensional, Lindelöf space that is not strongly D Open
A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\{U_x:x\in X\}$, there is a $D\subseteq X$ such that $\{U_x:x\in D\}$ covers $X$ and $D$ is locally finite in the topology generated by $\{U_x:x\in X\}$. We prove…
View article: Infinite monochromatic sumsets for colourings of the reals
Infinite monochromatic sumsets for colourings of the reals Open
N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of R so that no infinite sumset X + X is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we…
View article: INFINITE COMBINATORICS PLAIN AND SIMPLE
INFINITE COMBINATORICS PLAIN AND SIMPLE Open
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already…
View article: Ladder system uniformization on trees I & II
Ladder system uniformization on trees I & II Open
Given a tree $T$ of height $\omega_1$, we say that a ladder system colouring $(f_\alpha)_{\alpha\in \lim\omega_1}$ has a $T$-uniformization if there is a function $\varphi$ defined on a subtree $S$ of $T$ so that for any $s\in S_\alpha$ of…
View article: Ladder system uniformization on trees I & II
Ladder system uniformization on trees I & II Open
Given a tree $T$ of height $ω_1$, we say that a ladder system colouring $(f_α)_{α\in \limω_1}$ has a $T$-uniformization if there is a function $φ$ defined on a subtree $S$ of $T$ so that for any $s\in S_α$ of limit height and almost all $ξ…
View article: Two infinite quantities and their surprising relationship
Two infinite quantities and their surprising relationship Open
As early as the 17th century, Galileo Galilei wondered how to compare the sizes of infinite sets. Fast forward almost four hundred years, and in the summer of 2017, at the 6th European Set Theory Conference, a young model theorist, Maryant…
View article: A model with Suslin trees but no minimal uncountable linear orders other than $\omega_1$ and $-\omega_1$
A model with Suslin trees but no minimal uncountable linear orders other than $\omega_1$ and $-\omega_1$ Open
We show that the existence of a Suslin tree does not necessarily imply that there are uncountable minimal linear orders other than $\omega_1$ and $-\omega_1$, answering a question of J. Baumgartner. This is done by a Jensen-type iteration,…
View article: A model with Suslin trees but no minimal uncountable linear orders other than $ω_1$ and $-ω_1$
A model with Suslin trees but no minimal uncountable linear orders other than $ω_1$ and $-ω_1$ Open
We show that the existence of a Suslin tree does not necessarily imply that there are uncountable minimal linear orders other than $ω_1$ and $-ω_1$, answering a question of J. Baumgartner. This is done by a Jensen-type iteration, proving t…
View article: Orientations of graphs with uncountable chromatic number
Orientations of graphs with uncountable chromatic number Open
Motivated by an old conjecture of P. Erdős and V. Neumann‐Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable chromat…
View article: On spaces with $\sigma$-closed-discrete dense sets
On spaces with $\sigma$-closed-discrete dense sets Open
The main purpose of this paper is to study \\emph{$e$-separable spaces},\noriginally introduced by Kurepa as $K_0'$ spaces; we call a space $X$\n$e$-separable iff $X$ has a dense set which is the union of countably many\nclosed discrete se…
View article: On spaces with $σ$-closed-discrete dense sets
On spaces with $σ$-closed-discrete dense sets Open
The main purpose of this paper is to study \emph{$e$-separable spaces}, originally introduced by Kurepa as $K_0'$ spaces; we call a space $X$ $e$-separable iff $X$ has a dense set which is the union of countably many closed discrete sets. …
View article: Partial independent transversals in graphs avoiding large cliques
Partial independent transversals in graphs avoiding large cliques Open
Our goal is to investigate the independent transversal problem in the class of $K_n$-free graphs: we show that for any infinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal $r=r(G,m)$ so that for any balanced $r$-colo…