arXiv subjects

Lajos Soukup

Publications and source records attributed to Lajos Soukup.

At least 55 records · Page 3Linked to original sources

arXiv2017

Resolvability in c.c.c. generic extensions

Every crowded space $X$ is $ω$-resolvable in the c.c.c generic extension $V^{Fn(|X|,2})$ of the ground model. We investigate what we can say about $λ$-resolvability in c.c.c-generic extensions for $λ>ω$? A topological space is "monotonically $ω_1$-resolvable" if there is a function $f:X\to {ω_1}$ such that $$\{x\in X: f(x)\ge α \}\subset^{dense}X $$ for each $α<{ω_1}$. We show that given a $T_1$ space $X$ the following statements are equivalent: (1) $X$ is $ω_1$-resolvable in some c.c.c-generic extension, (2) $X$ is monotonically $ω_1$-resolvable. (3) $X$ is $ω_1$-resolvable in the Cohen-generic extension $V^{Fn({ω_1},2)}$. We investigate which spaces are monotonically $ω_1$-resolvable. We show that if a topological space $X$ is c.c.c, and $ω_1\le Δ(X)\le |X|<ω_ω$, then $X$ is monotonically $ω_1$-resolvable. On the other hand, it is also consistent, modulo the existence of a measurable cardinal, that there is a space $Y$ with $|Y|=Δ(Y)=\aleph_ω$ which is not monotonically $ω_1$-resolvable. The characterization of ${ω_1}$-resolvability in c.c.c generic extension raises the following question: is it true that crowded spaces from the ground model are $ω$-resolvable in $V^{Fn(ω,2)}$? We show that (i) if $V=L$ then every crowded c.c.c. space $X$ is $ω$-resolvable in $V^{Fn(ω,2)}$, (ii) if there is no weakly inaccesssible cardinals, then every crowded space $X$ is $ω$-resolvable in $V^{Fn(ω_1,2)}$. On the other hand, it is also consistent that there is a crowded space $X$ with $|X|=Δ(X)={ω_1}$ such that $X$ remains irresolvable after adding a single Cohen real.

math.GN↗

arXiv2016

First countable and almost discretely Lindelöf $T_3$ spaces have cardinality at most continuum

A topological space $X$ is called almost discretely Lindelöf if every discrete set $D \subset X$ is included in a Lindelöf subspace of $X$. We say that the space $X$ is {\em $μ$-sequential} if for every non-closed set $A \subset X$ there is a sequence of length $\le μ$ in $A$ that converges to a point which is not in $A$. With the help of a technical theorem that involves elementary submodels, we establish the following two results concerning such spaces. (1) For every almost discretely Lindelöf $T_3$ space $X$ we have $|X| \le 2^{χ(X)}$. (2) If $X$ is a $μ$-sequential $T_2$ space of pseudocharacter $ψ(X) \le 2^μ$ and for every free set $D \subset X$ we have $L(\overline{D}) \le μ$, then $|X| \le 2^μ$. The case $χ(X) = ω$ of (1) provides a solution to Problem 4.5 from "I. Juhász, V. Tkachuk, and R. Wilson, Weakly linearly Lindelöf monotonically normal spaces are Lindelöf", while the case $μ= ω$ of (2) is a partial improvement on the main result of "A.V. Archangel'skii and R.Z. Buzyakova, On some properties of linearly Lindelöf spaces".

math.GN↗

arXiv2015

Anti-Urysohn spaces

All spaces are assumed to be infinite Hausdorff spaces. We call a space "anti-Urysohn" $($AU in short$)$ iff any two non-emty regular closed sets in it intersect. We prove that $\bullet$ for every infinite cardinal $κ$ there is a space of size $κ$ in which fewer than $cf(κ)$ many non-empty regular closed sets always intersect; $\bullet$ there is a locally countable AU space of size $κ$ iff $ω\le κ\le 2^{\mathfrak c}$. A space with at least two non-isolated points is called "strongly anti-Urysohn" $($SAU in short$)$ iff any two infinite closed sets in it intersect. We prove that $\bullet$ if $X$ is any SAU space then $ \mathfrak s\le |X|\le 2^{2^{\mathfrak c}}$; $\bullet$ if $\mathfrak r=\mathfrak c$ then there is a separable, crowded, locally countable, SAU space of cardinality $\mathfrak c$; \item if $λ> ω$ Cohen reals are added to any ground model then in the extension there are SAU spaces of size $κ$ for all $κ\in [ω_1,λ]$; $\bullet$ if GCH holds and $κ\leλ$ are uncountable regular cardinals then in some CCC generic extension we have $\mathfrak s=κ$, $\,\mathfrak c=λ$, and for every cardinal $μ\in [\mathfrak s, \mathfrak c]$ there is an SAU space of cardinality $μ$. The questions if SAU spaces exist in ZFC or if SAU spaces of cardinality $> \mathfrak c$ can exist remain open.

math.GN↗

arXiv2015

Pinning Down versus Density

The pinning down number $ {pd}(X)$ of a topological space $X$ is the smallest cardinal $κ$ such that for any neighborhood assignment $U:X\to τ_X$ there is a set $A\in [X]^κ$ with $A\cap U(x)\ne\emptyset$ for all $x\in X$. Clearly, c$(X) \le {pd}(X) \le {d}(X)$. Here we prove that the following statements are equivalent: (1) $2^κ<κ^{+ω}$ for each cardinal $κ$; (2) ${d}(X)={pd}(X)$ for each Hausdorff space $X$; (3) ${d}(X)={pd}(X)$ for each 0-dimensional Hausdorff space $X$. This answers two questions of Banakh and Ravsky. The dispersion character $Δ(X)$ of a space $X$ is the smallest cardinality of a non-empty open subset of $X$. We also show that if ${pd}(X)<{d}(X)$ then $X$ has an open subspace $Y$ with ${pd}(Y)<{d}(Y)$ and $|Y| = Δ(Y)$, moreover the following three statements are equiconsistent: (i) There is a singular cardinal $λ$ with $pp(λ)>λ^+$, i.e. Shelah's Strong Hypothesis fails; (ii) there is a 0-dimensional Hausdorff space $X$ such that $|X|=Δ(X)$ is a regular cardinal and ${pd}(X)<{d}(X)$; (iii) there is a topological space $X$ such that $|X|=Δ(X)$ is a regular cardinal and ${pd}(X)<{d}(X)$. We also prove that $\bullet$ ${d}(X)={pd}(X)$ for any locally compact Hausdorff space $X$; $\bullet$ for every Hausdorff space $X$ we have $|X|\le 2^{2^{{pd}(X)}}$ and ${pd}(X)<{d}(X)$ implies $Δ(X)< 2^{2^{{pd}(X)}}$; $\bullet$ for every regular space $X$ we have $\min\{Δ(X),\, w(X)\}\le 2^{{pd}(X)}\,$ and ${d}(X)<2^{{pd}(X)},\,$ moreover ${pd}(X)<{d}(X)$ implies $\,Δ(X)< {2^{{pd}(X)}}$.

math.GN↗

arXiv2014

Between countably compact and $ω$-bounded

Given a property $P$ of subspaces of a $T_1$ space $X$, we say that $X$ is {\em $P$-bounded} iff every subspace of $X$ with property $P$ has compact closure in $X$. Here we study $P$-bounded spaces for the properties $P \in \{ωD, ωN, C_2 \}$ where $ωD \, \equiv$ "countable discrete", $ωN \, \equiv$ "countable nowhere dense", and $C_2 \,\equiv$ "second countable". Clearly, for each of these $P$-bounded is between countably compact and $ω$-bounded. We give examples in ZFC that separate all these boundedness properties and their appropriate combinations. Consistent separating examples with better properties (such as: smaller cardinality or weight, local compactness, first countability) are also produced. We have interesting results concerning $ωD$-bounded spaces which show that $ωD$-boundedness is much stronger than countable compactness: $\bullet$ Regular $ωD$-bounded spaces of Lindelöf degree $< cov(\mathcal{M})$ are $ω$-bounded. $\bullet$ Regular $ωD$-bounded spaces of countable tightness are $ωN$-bounded, and if $\mathfrak{b} > ω_1$ then even $ω$-bounded. $\bullet$ If a product of Hausdorff space is $ωD$-bounded then all but one of its factors must be $ω$-bounded. $\bullet$ Any product of at most $\mathfrak{t}$ many Hausdorff $ωD$-bounded spaces is countably compact. As a byproduct we obtain that regular, countably tight, and countably compact spaces are discretely generated.

math.GN↗

arXiv2014

Partitioning bases of topological spaces

We investigate whether an arbitrary base for a dense-in-itself topological space can be partitioned into two bases. We prove that every base for a T_3 Lindelöf topology can be partitioned into two bases while there exists a consistent example of a first countable, 0-dimensional, Hausdorff space of size continuum and weight ω_1 which admits a point countable base without a partition to two bases. Several related results are proved and the paper finishes with a list of open problems.

math.GN↗

arXiv2013

Constructing, sampling and counting graphical realizations of restricted degree sequences

With the current burst of network theory (especially in connection with social and biological networks) there is a renewed interest on realizations of given degree sequences. In this paper we propose an essentially new degree sequence problem: we want to find graphical realizations of a given degree sequence on labeled vertices, where certain would-be edges are {\em forbidden}. Then we want to sample uniformly and efficiently all these possible realizations. (This problem can be considered as a special case of Tutte's $f$-factor problem, however it has a favorable sampling speed.) We solve this {\em restricted degree sequence} (or RDS for short) problem completely if the forbidden edges form a bipartite graph, which consist of the union of a (not necessarily maximal) 1-factor and a (possible empty) star. Then we show how one can sample the space of all realizations of these RDSs uniformly and efficiently when the degree sequence describes a {\em half-regular} bipartite graph. Our result contains, as special cases, the well-known result of Kannan, Tetali and Vempala on sampling regular bipartite graphs and a recent result of Greenhill on sampling regular directed graphs (so it also provides new proofs of them). The RDS problem descried above is self-reducible, therefore our {\em fully polynomial almost uniform sampler} (a.k.a. FPAUS) on the space of all realizations also provides a {\em fully polynomial randomized approximation scheme} (a.k.a. FPRAS) for approximate counting of all realizations.

math.CO↗

arXiv2013

Regular spaces of small extent are omega-resolvable

We improve some results of Pavlov and of Filatova, respectively, concerning a problem of Malychin by showing that every regular space X that satisfies Delta(X)>ext(X) is omega-resolvable. Here Delta(X), the dispersion character of X, is the smallest size of a non-empty open set in X and ext(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are omega-resolvable. We also prove that any regular Lindelöf space X with |X|=Δ(X)=omega_1 is even omega_1-resolvable. The question if regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.

math.GN↗

arXiv2012

Comparing weak versions of separability

Our aim is to investigate spaces with sigma-discrete and meager dense sets, as well as selective versions of these properties. We construct numerous examples to point out the differences between these classes while answering questions of Tkachuk [30], Hutchinson [17] and the authors of [8].

math.GN↗

arXiv2012

Towards random uniform sampling of bipartite graphs with given degree sequence

In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on $n$ vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in $n$ in case of {\em semi-regular} degree sequence. The novelty of our approach lays in the construction of the canonical paths in Sinclair's method.

math.CO↗

arXiv2010

Elementary submodels in infinite combinatorics

The usage of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williams's theorem on cycle-decomposition of graphs, and finally we improve a decomposition theorem of Laviolette concerning bond-faithful decompositions of graphs.

math.LO↗

arXiv2010

Wide scattered spaces and morasses

We show that it is relatively consistent with ZFC that 2^omega is arbitrarily large and every sequence s=(s_i:i<omega_2) of infinite cardinals with s_i<=2^omega is the cardinal sequence of some locally compact scattered space.

math.LO↗

arXiv2010

Pcf theory and cardinal invariants of the reals

The additivity spectrum ADD(I) of an ideal I is the set of all regular cardinals kappa such that there is an increasing chain {A_alpha:alpha<kappa\} in the ideal I such that the union of the chain is not in I. We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that I=B or I=N, where B denotes the sigma-ideal generated by the compact subsets of the Baire space omega^omega, and N is the ideal of the null sets. For countable sets we give a full characterization of the additivity spectrum of I: a non-empty countable set A of uncountable regular cardinals can be ADD(I) in some c.c.c generic extension iff A=pcf(A).

math.LO↗

arXiv2010

Superatomic Boolean algebras constructed from strongly unbounded functions

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $κ,λ$ are infinite cardinals such that $κ^{+++} \leq λ$, $κ^{<κ}=κ$ and $2^κ= κ^+$, and $η$ is an ordinal with $κ^+\leq η<κ^{++}$ and $cf(η) = κ^+$. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra $B$ such that - $ht(B) = η+ 1$, - the cardinality of the $α$th level of $B$ is $κ$ for every $α<η$, - and the cardinality of the $η$th level of $B$ is $λ$ Especially, $\<ω\>_{ω_1}\concatenation \<ω_3\>$ and $\<ω_1\>_{ω_2}\concatenation \<ω_4\>$ can be cardinal sequences of superatomic Boolean algebras.

math.LO↗

arXiv2010

Balanced Vertices in Trees and a Simpler Algorithm to Compute the Genomic Distance

This paper provides a short and transparent solution for the covering cost of white-grey trees which play a crucial role in the algorithm of Bergeron {\it et al.}\ to compute the rearrangement distance between two multichromosomal genomes in linear time ({\it Theor. Comput. Sci.}, 410:5300-5316, 2009). In the process it introduces a new {\em center} notion for trees, which seems to be interesting on its own.

cs.DM↗

arXiv2010

Conflict free colorings of (strongly) almost disjoint set-systems

A set-system $X$ is a $(λ, κ,μ)$-system iff $|X|=λ$, $|x|=κ$ for each $x\in X$, and $X$ is $μ$-almost disjoint. We write $[λ, κ, μ] -> ρ$ iff every $(λ, κ,μ)$-system has a "conflict free coloring with $ρ$ colors", i.e. there is a coloring of the elements of $\cup X$ with$ρ$ colors such that for each element $x$ of $X$ there is a color $ξ<ρ$ such that exactly one element of $x$ has color $ξ$. Our main object of study is the relation $[λ, κ, μ] -> ρ$. We give full description of this relation when $κ$ is finite. We also show that if $d$ is a natural number then $[λ,κ,d]-> ω$ always holds. Under GCH we prove that $[λ,κ,ω]-> ω_2$ holds for $κ>ω_1$, but the relation $[λ,κ,ω]-> ω_1$ is independent (modulo some large cardinals).

math.LO↗

arXiv2010

Club guessing for dummies

We give a direct, detailed and relatively short proof of Shelah's theorem on club guessing sequences on $S^{μ^+}_μ$ (for any regular, uncountable cardinal $μ$).

math.LO↗

arXiv2010

A note on Noetherian type of spaces

The Noetherian type of a space X, Nt(X), is the least cardinal kappa such that X has a base B such that every element of the base is contained in less than kappa many elements of the base. Denote X the space obtained from 2^{aleph_omega} by declaring the G_delta sets to be open. Milovich proved that if Square_{aleph_omega} holds and (aleph_omega)^omega=aleph_{omega+1} then Nt(X)=omega_1. Answering a question of Spadaro, we show that if (aleph_omega)^omega=aleph_{omega+1} and a strong form of Chang Conjecture holds for aleph_ωthen Nt(X)>omega_1.

math.GN↗