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Dániel T. Soukup

Publications and source records attributed to Dániel T. Soukup.

13 recordsLinked to original sources
arXiv2020

Reducing the dichromatic number via cycle reversions in infinite digraphs

We prove the following conjecture of S. Thomassé: for every (potentially infinite) digraph $ D $ it is possible to iteratively reverse directed cycles in such a way that the dichromatic number of the final reorientation $ D^{*} $ of $ D $ is at most two and each edge is flipped only finitely many times. In addition, we guarantee that in every strong component of $ D^{*} $ all the local edge-connectivities are finite and any edge is reversed at most twice.

math.CO↗
arXiv2020

Extremal triangle-free and odd-cycle-free colourings of uncountable graphs

The optimality of the Erdős-Rado theorem for pairs is witnessed by the colouring $Δ_κ: [2^κ]^2 \rightarrow κ$ recording the least point of disagreement between two functions. This colouring has no monochromatic triangles or, more generally, odd cycles. We investigate a number of questions investigating the extent to which $Δ_κ$ is an \emph{extremal} such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of $Δ$-regressive and almost $Δ$-regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether $Δ_κ$ has the minimal cardinality of any \emph{maximal} triangle-free or odd-cycle-free colouring into $κ$. We resolve the question positively for odd-cycle-free colourings.

math.LO↗
arXiv2019

Towers and gaps at uncountable cardinals

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 $\mathfrak p(κ)=\mathfrak t(κ)$ or there is a $(\mathfrak p(κ),λ)$-gap of club-supported slaloms for some $λ< \mathfrak p(κ)$. While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah's proof of $\mathfrak p=\mathfrak t$ to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that $\mathfrak p(κ)$ is always regular; the latter extends results of Garti. Finally, we turn to club variants of $\mathfrak p(κ)$ and present a new model for the inequality $\mathfrak{p}(κ) = κ^+ < \mathfrak{p}_{cl}(κ) = 2^κ$. In contrast to earlier arguments by Shelah and Spasojevic, we achieve this by adding $κ$-Cohen reals and then successively diagonalising the club-filter; the latter is shown to preserve a Cohen witness to $\mathfrak{p}(κ) = κ^+$.

math.LO↗
arXiv2019

On the complexity of classes of uncountable structures: trees on $\aleph_1$

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 are empty). Moreover, under $(V=L)$, (a) the class of Aronszajn and Suslin trees is $Π_1^1$-complete, (b) the class of special Aronszajn trees is $Σ_1^1$-complete, and (c) the class of Kurepa trees is $Π^1_2$-complete. We achieve these results by finding nicely definable reductions that map subsets $X$ of $ω_1$ to trees $T_X$ so that $T_X$ is in a given tree-class $\mathcal T$ if and only if $X$ is stationary/non-stationary (depending on the class $\mathcal T$). Finally, we present models of CH where these classes have lower projective complexity.

math.LO↗
arXiv2019

Ladder system uniformization on trees I & II

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 $ξ\in {dom} (f_α)$, $φ(s\upharpoonright ξ)=f_α(ξ)$. In sharp contrast to the classical theory of uniformizations on $ω_1$, J. Moore proved that CH is consistent with the statement that any ladder system colouring has a $T$-uniformization (for any Aronszajn tree $T$). Our goal is to present a fine analysis of ladder system uniformization on trees pointing out the analogies and differences between the classical and this new theory. We show that if $S$ is a Suslin tree then (i) CH implies that there is a ladder system colouring without $S$-uniformization; (ii) the restricted forcing axiom $MA(S)$ implies that any ladder system colouring has an $ω_1$-uniformization. For an arbitrary Aronszajn tree $T$, we show how diamond-type assumptions affect the existence of ladder system colourings without a $T$-uniformization. Furthermore, it is consistent that for any Aronszajn tree $T$ and ladder system $\mathbf C$ there is a colouring of $\mathbf C$ without a $T$-uniformization; however, and quite surprisingly, $\diamondsuit^+$ implies that for any ladder system $\mathbf C$ there is an Aronszajn tree $T$ so that any monochromatic colouring of $\mathbf C$ has a $T$-uniformization. We also prove positive uniformization results in ZFC for some well-studied trees of size continuum, and finish with a list of open problems.

math.LO↗
arXiv2019

Infinite monochromatic sumsets for colourings of the reals

N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $\mathbb R$ so that no infinite sumset $X+X=\{x+y:x,y\in X\}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:\mathbb R\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb R$ so that $c$ is constant on $X+X$.

math.LO↗
arXiv2018

Two infinite quantities and their surprising relationship

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, Maryanthe Malliaris, and the well-known polymath, Saharon Shelah, received the Hausdorff Medal for the most influential work in set theory published in the last five years. Malliaris and Shelah made significant breakthroughs both regarding a model theoretic classification problem (that is, sorting certain objects into types), and proved that two well-studied infinite quantities, $\mathfrak{p}$ and $\mathfrak{t}$, are in fact the same. This latter result is the focus of our expository paper.

math.HO↗
arXiv2018

The open dihypergraph dichotomy and the second level of the Borel hierarchy

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 perfect set theorem. Under the axiom of determinacy, we obtain the generalizations of these results from analytic metric spaces to separable metric spaces. We also consider connections between cardinal invariants and the chromatic numbers of the corresponding dihypergraphs.

math.LO↗
arXiv2018

Infinite combinatorics plain and simple

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, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

math.LO↗
arXiv2018

Uncountable strongly surjective linear orders

A linear order $L$ is strongly surjective if $L$ can be mapped onto any of its suborders in an order preserving way. We prove various results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of Camerlo, Carroy and Marcone. In particular, $\diamondsuit^+$ implies the existence of a lexicographically ordered Suslin-tree which is strongly surjective and minimal; every strongly surjective linear order must be an Aronszajn type under $2^{\aleph_0}<2^{\aleph_1}$ or in the Cohen and other canonical models (where $2^{\aleph_0}=2^{\aleph_1}$); finally, we prove that it is consistent with CH that there are no uncountable strongly surjective linear orders at all. We end the paper with a healthy list of open problems.

math.LO↗
arXiv2017

Orientations of graphs with uncountable chromatic number

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 chromatic number if its vertices cannot be covered by countably many independent sets, and a digraph has uncountable dichromatic number if its vertices cannot be covered by countably many acyclic sets. We prove that consistently there are digraphs with uncountable dichromatic number and arbitrarily large digirth; this is in surprising contrast with the undirected case: any graph with uncountable chromatic number contains a 4-cycle. Next, we prove that several well known graphs (uncountable complete graphs, certain comparability graphs, and shift graphs) admit orientations with uncountable dichromatic number in ZFC. However, we show that the statement "every graph $G$ of size and chromatic number $ω_1$ has an orientation $D$ with uncountable dichromatic number" is independent of ZFC.

math.CO↗
arXiv2014

Trees, ladders and graphs

We introduce a new method to construct uncountably chromatic graphs from non special trees and ladder systems. Answering a question of P. Erdős and A. Hajnal from 1985, we construct graphs of chromatic number $ω_1$ without uncountable $ω$-connected subgraphs. Second, we build triangle free graphs of chromatic number $ω_1$ without subgraphs isomorphic to $H_{ω,ω+2}$.

math.CO↗