arXiv subjects

Lajos Soukup

Publications and source records attributed to Lajos Soukup.

At least 19 recordsLinked to original sources

arXiv2026

Stacking and Clearing in Directed Graph Pebbling

Suppose that pebbles are distributed on the vertices of a directed graph D. A directed pebbling step u -> v along an arc u -> v removes two pebbles from u and places one pebble on v. We study the stacking number stack(D), the least integer t >= 2 such that every configuration with t pebbles can be transformed by a finite sequence of pebbling steps into a configuration with all pebbles on a single vertex, and the clearing number clear(D), defined analogously by requiring a final configuration with one pebble. Our main result is the formula stack(C_n) = n(2^{n-1}-1)+1 for the directed n-cycle C_n, for n >= 2. We also prove that, for finite simple digraphs with at least two vertices, stack(D) is defined precisely for strongly connected digraphs, and clear(D) is defined precisely for strongly connected digraphs whose directed cycle lengths have greatest common divisor 1.

math.CO↗

arXiv2026

Stacking and clearing in graph pebbling

Suppose that pebbles are distributed on the vertices of a graph G. A pebbling step along an edge uv removes two pebbles from u and places one pebble on v. We introduce two new graph parameters: stack(G): the least integer t such that every configuration with t pebbles can be transformed, by a finite sequence of pebbling steps, into a configuration with all pebbles on a single vertex. clear(G): defined analogously, but requiring that from every configuration with t pebbles, all but one pebble can be removed. We prove that stack(G) is defined exactly for connected graphs, and that clear(G) is defined exactly for connected non-bipartite graphs. We also establish general upper bounds for these parameters; in particular, stack(G), clear(G) <= 2 |V(G)| 2^diam(G), where diam(G) denotes the diameter of G. Among our exact results are the equalities stack(K_n) = clear(K_n) = n + 1, stack(K_{m,n}) = 3 max{m,n} + 1, stack(P_n) = 2^n - 1. We also establish general lower bounds in terms of the independence number and odd closed walks. For cycles, the situation is more delicate. We prove the lower bounds stack(C_{2n}) >= 2^{n+1} - 1, clear(C_{2n+1}) >= 3 * 2^n - 2, and formulate the Almost Stacked Hypothesis, motivated by Sjostrand's cover pebbling theorem. Assuming this hypothesis, we obtain stack(C_{2n}) = 2^{n+1} - 1, clear(C_{2n+1}) = 3 * 2^n - 2. At present, we do not have a conjecture for the exact value of stack(C_{2n+1}). Finally, computational evidence leads us to a conjectural closed formula for the stacking number of a tree in terms of the distances and degrees of the vertices relative to a chosen root.

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arXiv2025

Noetherian Properties, Large Cardinals, and Independence Around $\aleph_{\omega}$

A base of a topological space is called {\em Noetherian } iff it does not contain an infinite strictly $\subseteq$-increasing chain. We show that minimal cardinality of a regular spaces without a Noetherian base is the first strongly inaccessible cardinal, answering a question from the 1980s. We also study the {\em Noetherian type} of a topological space $X$, denoted by $Nt(X)$, defined as the least cardinal $\kappa$ such that $X$ has a base $\mathcal B$ with $|\{B'\in \mathcal B: B\subset B'\}|<\kappa$ for each $B\in \mathcal B$. The behavior of the Noetherian type under the $G_\delta$-modification was investigated by Milovich and Spadaro. A central question, posed by them, is whether the Noetherian type of the $G_{\delta}$-modification of the space $D(2)^{\aleph_\omega}$ is $\omega_1$. This statement, denoted (Nt), is known to be independent of ZFC + GCH: it holds under ``GCH + $\square_{\aleph_\omega}$'', but fails under ``GCH + $(\aleph_{\omega+1}, \aleph_\omega)\to (\aleph_1, \aleph_0)$''. We place this phenomenon in a broader context by identifying similar independence phenomena for several topological and combinatorial principles. These include: (wFN) the weak Freese-Nation property of $[\aleph_{\omega}]^{\omega}$; (SAT) the existence of a saturated MAD family in $[\aleph_{\omega}]^{\omega}$; (HnT) the existence of an ${\omega}$-homogeneous, but not ${\omega}$-transitive permutation group on $\aleph_{\omega}$; and (SPL) the existence of a countably compact, locally countable, and ${\omega}$-fair regular space of cardinality $\aleph_{\omega+1}$. Assuming GCH, we analyze the logical relationships between these principles and show, for example, that SPL implies wFN, which in turn implies both SAT, HnT and Nt, while SAT does not imply Nt.

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arXiv2025

A consistency theorem for cardinal sequences of length $< \omega_3$

We prove that if $\lambda$ is a fixed uncountable cardinal and $f = \langle \ka_{\al} : \al < \delta \rangle$ is a sequence of infinite cardinals where $\delta < \omega_3$ and $\ka_{\al}\in \{\om,\lambda\}$ for each $\al < \delta$ in such a way that $f^{-1}\{\om\}$ is $\om_2$-closed in $\delta$, then it is consistent that there is a scattered Boolean space whose cardinal sequence is $f$.

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arXiv2025

Any fully graphic region of degree sequences can be sampled rapidly

Let $n>c_1\ge c_2$ and $\Sigma$ be positive integers with $n\cdot c_1\ge \Sigma \ge n\cdot c_2.$ Let $\mD=\dds{n}{\Sigma}{c_1}{c_2}$ denote the set of all degree sequences of length $n$ with the even sum $\Sigma$ and satisfying $c_1\ge d_i\ge c_2.$ We show that if all degree sequences in $\mD$ are graphic, then $\mD$ is $3n^{13}$-stable. (The concept of $P$-stability was introduced by Jerrum and Sinclair in 1990.) In particular, this implies that the switch Markov-chain mixes rapidly on all such degree sequences. In this paper we also study the inverse direction. We show the following: if all graphic sequences of a degree sequence region satisfy the $p(n)$-stability condition then the overwhelming majority of the sequences in the region is graphic. This answers affirmatively a question raised in the paper \DOI{10.1016/j.aam.2024.102805}.

math.CO↗

arXiv2025

Condensations with extra properties

We show that there are locally compact spaces that can be condensed onto separable spaces but not onto compact separable spaces. We also show that for every cardinal $\kappa$ there is a locally compact topological group of cardinality $2^\kappa$ that can be condensed onto a compact space but not onto a compact topological group. These answer some questions of Arhangel'skii and Buzyakova.

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arXiv2025

Cut-and-choose games in topological spaces

We study transfinite cut-and-choose games on $T_0$ spaces, introducing the {\em point-separating number} $ps(X)$ and the {\em set membership number} ${sm}(X)$ as the ordinal-valued invariants measuring the minimal length of a game in which a Seeker can determine a hidden point or subset. A central motivating question is which countable ordinals can occur as the value of $ps(X)$, in particular whether any countable ordinal can arise. These invariants generalize Scott's $T_0$-pseudoweight $\psi w_0$. We establish fundamental inequalities relating $ps(X)$, ${sm}(X)$, $\psi w_0(X)$, and $|X|$, including the sharp bounds $|X|\le 2^{ps(X)}$ and $\psi w_0(X)\le 2^{<ps(X)}$. We compute these invariants for familiar spaces such as Cantor cubes, powers of the Alexandroff double arrow space, and certain stationary subsets of cardinals. We further investigate their behavior under topological sums and products, revealing the striking contrast between $ps$ and ${sm}$. For metric spaces, we determine that $ps(X)=\log|X|$. However, we do not know such computation for ${sm}(X)$; we can only assert that ${sm}(X)$ may be arbitrarily large. Finally, we highlight another open problem: whether these games are always determined.

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arXiv2024

Fully graphic degree sequences and P-stable degree sequences

The notion of $P$-stability of an infinite set of degree sequences plays influential role in approximating the permanents, rapidly sampling the realizations of graphic degree sequences, or even studying and improving network privacy. While there exist several known sufficient conditions for $P$-stability, we don't know any useful necessary condition for it. We also do not have good insight of possible structure of $P$-stable degree sequence families. At first we will show that every known infinite $P$-stable degree sequence set, described by inequalities of the parameters $n, c_1, c_2, \Sigma$ (the sequence length, the maximum and minimum degrees and the sum of the degrees) is ,,fully graphic" meaning that every degree sequence from the region with an even degree sum, is graphic. Furthermore, if $\Sigma$ does not occur in the determining inequality, then the notions of $P$-stability and full graphicality will be proved equivalent. In turns, this equality provides a strengthening of the well-known theorem of Jerrum, McKay and Sinclair about $P$-stability, describing the maximal $P$-stable sequence set by $n, c_1, c_2$. Furthermore we conjecture that similar equivalences occur in cases if $\Sigma$ also part of the defining inequality.

math.CO↗

arXiv2024

The class $C(ω_1)$ and countable net weight

Hart and Kunen, and independently in the recent preprint arXiv:2304.13113, Ríos-Herrejón defined and studied the class $C(ω_1)$ of topological spaces $X$ having the property that for every neighborhood assignment $\{U(y) : y \in Y\}$ with $Y \in [X]^{ω_1}$ there is $Z \in [Y]^{ω_1}$ such that $$Z \subset \bigcap \{U(z) : z \in Z\}.$$ It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these and several other classes that are sandwiched between them. These clarify some of the main problems that were raised in the above preprint. In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in $C(ω_1)$ has countable net weight, answering a question that was raised by Hart and Kunen.

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arXiv2024

Selectively pseudocompact spaces

A novel selection principle was introduced by Dorantes-Aldama and Shakhmatov: a topological space $X$ is termed {\em selectively pseudocompact} if for any sequence $(U_n:n\in {\omega})$ of pairwise disjoint non-empty open sets of $X$, one can choose points $x_n\in U_n$ such that the sequence $(x_n:n\in {\omega})$ has an accumulation point. In this paper, we explore various versions of this principle when we permit the selection of finite, scattered, or nowhere dense sets instead of just singletons. We develop a method to prove that the aforementioned versions of selective pseudocompactness are indeed distinct from one another.

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arXiv2024

On the companion of spaces having dense, relatively countable compact subspaces

The notion of "pseudocompactness" was introduced by Hewitt. The concept of relatively countably compact subspaces were explored by Marjanovic to show that a $Ψ$-space is pseudocompact. A topological space is said to be DRC (DRS) iff it possesses a dense, relatively countably compact (or relatively sequentially compact, respectively) subspace. The concept of selectively pseudocompact game Sp(X) and the selectively sequentially pseudocompact game Ssp(X) were introduced by Dorantes-Aldama and Shakhmatov. They explored the relationship between the existence of a winning strategy and a stationary winning strategy for player P in these games. In particular, they observed that there exists a stationary winning strategy in the game Sp(X) (Ssp(X)) for Player P iff $X$ is DRC (or DRS, respectively). In this paper we introduce natural weakening of the properties DRC and DRS: a space $X$ is DRCo ( DRSo) iff there is a sequence $(D_n:n \in { ω})$ of dense subsets of $X$ such that every sequence $(d_n:n \in { ω} )$ with $d_n \in D_n$ has an accumulation point (or contains a convergent subsequence, respectively). These properties are also equivalent to the existence of some limited knowledge winning strategy on the corresponding games $Sp(X)$ and $Ssp(X)$. Clearly, DRS implies DRC and DRSo, DRC or DRSo imply DRCo. The main part of this paper is devoted to prove that apart from these trivial implications, consistently there are no other implications between these properties.

math.GN↗

arXiv2023

Infinite Combinatorics revisited in the absence of Axiom of Choice

We investigate the provability of classical combinatorial theorems in ZF. Using combinatorial arguments, we establish the following results for each infinite cardinal $κ\in On$, (1) $κ^+\to (κ,ω+1)$, (2) any family $\mathcal A\subset [{On}]^{<ω}$ of size $κ^+$ contains a $Δ$-system of size $κ$, (3) given a set mapping $F:κ\to {[κ]}^{<ω}$, the set $κ$ has a partition into $ω$-many $F$-free sets, By employing Karagila's method of absoluteness, we prove the following for each uncountable cardinal $κ\in On$, (4) given a set mapping $F:κ\to {[κ^]}^{<ω}$, there is an $F$-free set of cardinality $κ$, (5) for each natural number $n$, every family $\mathcal A\subset {[κ]}^{ω}$with $|A\cap B|\le n$ for $\{A,B\}\in {[\mathcal A]}^{2}$ has property $B$, In contrast to (5), we show that the following statement is not provable from ZF + $cf(ω_1)=ω_1$: (6*) every family $\mathcal A\subset {[ω_1]}^{ω}$ with $|A\cap B|\le 1$ for $\{A,B\}\in {[\mathcal A]}^{2}$ is "essentially disjoint" . The following statements are not provable in ZF, but they are equivalent in ZF: (i) $cf(ω_1)=ω_1$, (ii) $ω_1\to (ω_1,ω+1)^2$, (iii) any family $\mathcal A\subset [{On}]^{<ω}$ of size $ω_1$ contains a $Δ$-system of size $ω_1$. A function $f$ is a "uniform denumeration on $ω_1$" iff $dom(f)=ω_1$ and for every $α<ω_1$, $f(α)$ is a function from $ω$ onto $α$. It is evident that the existence of a uniform denumeration of $ω_1$ implies $cf(ω_1)=ω_1$. We prove that the failure of the reverse implication is equiconsistent with the existence of an inaccessible cardinal.

math.LO↗

arXiv2022

Essentially disjoint families, conflict free colorings and Shelah's Revised GCH

Using Shelah's revised GCH theorem we prove that if mu =kappa for all b from B has a conflict-free colorings with kappa colors. Putting together these results we obtain that if mu =beth_omega for all b from B has a conflict-free colorings with beth_omega colors. To yield the above mentioned results we also need to prove a certain compactness theorem concerning singular cardinals.

math.LO↗

arXiv2022

Elusive properties of infinite graphs

A graph property is said to be elusive ( evasive) if every algorithm testing this property by asking questions of the form "is there an edge between vertices x and y" requires, in the worst case, to ask about all pairs of vertices. The unsettled Aanderaa-Karp-Rosenberg conjecture is that every monotone graph property is elusive for finite vertex sets. We show that the situation is completely different for infinite vertex sets: the monotone graph properties "every vertex has degree at least n" and "every connected components has size at least n" where n is a natural number, are not elusive for infinite vertex sets, but the monotone graph property "the graph contains a cycle" is elusive for arbitrary vertex sets. On the other hand, we also prove that every algorithm testing some natural monotone graph properties, e.g "every vertex has degree at least n" or "connected" on the vertex set omega should check "lots of edges", more precisely, all the edges of an infinite complete subgraph.

math.CO↗

arXiv2022

On resolvability of products

All spaces below are $T_0$ and crowded (i.e. have no isolated points). For $n \le ω$ let $M(n)$ be the statement that there are $n$ measurable cardinals and $Π(n)$ ($Π^+(n)$) that there are $n+1$ (0-dimensional $T_2$) spaces whose product is irresolvable. We prove that $M(1),\,Π(1)$ and $Π^+(1)$ are equiconsistent. For $1 < n < ω$ we show that $CON(M(n))$ implies $CON(Π^+(n))$. Finally, $CON(M(ω))$ implies the consistency of having infinitely many crowded 0-dimensional $T_2$-spaces such that the product of any finitely many of them is irresolvable. These settle old problems of Malychin. Concerning an even older question of Ceder and Pearson, we show that the following are consistent modulo a measurable cardinal: (i) There is a 0-dimensional $T_2$ space $X$ with $ω_2 \le Δ(X) \le 2^{ω_1}$ whose product with any countable space is not $ω_2$-resolvable, hence not maximally resolvable. (ii) There is a monotonically normal space $X$ with $Δ(X) = \aleph_ω$ whose product with any countable space is not $ω_1$-resolvable, hence not maximally resolvable. These significantly improve a result of Eckertson.

math.GN↗

arXiv2021

Constructions of Lindelöf scattered P-spaces

We construct locally Lindelöf scattered P-spaces (LLSP spaces, in short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width $ω_1$ and height $ω_2$ and that it is relatively consistent with ZFC that there is an LLSP space of width $ω_1$ and height $ω_3$. Also, we prove a stepping up theorem that, for every cardinal $λ\geq ω_2$, permits us to construct from an LLSP space of width $ω_1$ and height $λ$ satisfying certain additional properties an LLSP space of width $ω_1$ and height $α$ for every ordinal $α< λ^+$. Then, we obtain as consequences of the above results the following theorems: (1) For every ordinal $α< ω_3$ there is an LLSP space of width $ω_1$ and height $α$. (2) It is relatively consistent with ZFC that there is an LLSP space of width $ω_1$ and height $α$ for every ordinal $α< ω_4$.

math.LO↗

arXiv2021

On a problem of Angelo Bella

The main result of this note is the following theorem. "If $X$ is any Hausdorff space with $κ= \widehat{F}(X) \cdot \widehatμ(X)$ then $L(X_{< κ}) \le \varrho(κ)$". Here $\widehat{F}(X)$ is the smallest cardinal $φ$ so that $|S| < φ$ for any set $S$ that is free in $X$ and $\widehatμ(X)$ is the smallest cardinal $μ$ so that, for every set $S$ that is free in $X$, any open cover of $\overline {S}$ has a subcover of size $< μ$. Moreover, $X_{< κ}$ is the $G_{< κ}$-modification of $X$ and $\varrho(κ) = \min \{\varrho : \varrho ^{< κ} = \varrho\}$. As a corollary we obtain that if $X$ is a linearly Lindelöf regular space of countable tightness then $L(X_δ) \le \mathfrak{c}$, provided that $ \mathfrak{c} = 2^{< \mathfrak{c}}$. This yields a consistent affirmative answer to a question of Angelo Bella.

math.GN↗

arXiv2021

The double density spectrum of a topological space

It is an interesting, maybe surprising, fact that different dense subspaces of even "nice" topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological space $X$ that we call the double density spectrum of $X$ and denote by $dd(X)$. We improve a result of Berner and Juhasz by showing that $dd(X)$ is always $ω$-closed (i.e. countably closed) if $X$ is Hausdorff. We manage to give complete characterizations of the double density spectra of Hausdorff and of regular spaces as follows. Let $S$ be a non-empty set of infinite cardinals. Then (1) $S = dd(X)$ holds for a Hausdorff space $X$ iff S is $ω$-closed and $sup S \le 2^{2^{\min S}},$ (2) S = dd(X) holds for a regular space X iff S is $ω$-closed and $\sup S \le {2^{\min S}}$. We also prove a number of consistency results concerning the double density spectra of compact spaces. For instance: (i) If $κ= cf(κ)$ embeds in $\mathcal{P}(ω)/fin$ and $S$ is any set of uncountable regular cardinals $< κ$ with $|S| < \min S$, then there is a compactum $C$ such that $\{ω, κ\} \cup S \subset dd(C)$, moreover $λ\notin d(C)$ whenever $|S| + ω< cf(λ) < κ$ and $cf(λ) \notin S$. (ii) It is consistent to have a separable compactum $C$ such that $dd(C)$ is not $ω_1$-closed.

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