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Tamás Csernák

Publications and source records attributed to Tamás Csernák.

10 recordsLinked to original sources
arXiv2026

The Almost Stacked Hypothesis: A Conjectural Analogue of Sjöstrand's Cover Pebbling Theorem

We study two graph pebbling parameters, the stacking number and the clearing number, through the Almost Stacked Hypothesis (ASH). This hypothesis asserts that these thresholds can be determined by testing only configurations in which at most one vertex carries more than one pebble. We prove that every almost stacked configuration of size $2^{n+1}-1$ on $C_{2n}$ is stackable and that every almost stacked configuration of size $3\cdot 2^n-2$ on $C_{2n+1}$ is clearable. Together with the known lower bounds, these results show that ASH implies $\operatorname{stack}(C_{2n})=2^{n+1}-1$ and $\operatorname{clear}(C_{2n+1})=3\cdot 2^n-2$. For a finite tree $T$, we introduce an explicit invariant $\operatorname{estim}(T)$. We prove unconditionally that $\operatorname{stack}(T)\geq\operatorname{estim}(T)$ and prove the reverse inequality under ASH. Consequently, ASH yields $\operatorname{stack}(T)=\operatorname{estim}(T)$, and we conjecture that this equality holds unconditionally. Finally, we study perfectly pebblable graphs: finite connected non-bipartite graphs whose clearing number has the minimum possible value $\operatorname{clear}(G)=|V(G)|+1$. Every complete graph with at least three vertices is perfectly pebblable, which might suggest that perfect pebblability requires high edge density. Assuming ASH, however, we show that this is not the case. We give a sufficient criterion involving strong edge-triangulation and Hamiltonian-path and path-cover conditions in vertex-deleted subgraphs and use it to construct two explicit infinite families of perfectly pebblable graphs with edge density tending to zero, one of which has only a linear number of edges.

math.CO↗
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

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 $ψw_0$. We establish fundamental inequalities relating $ps(X)$, ${sm}(X)$, $ψw_0(X)$, and $|X|$, including the sharp bounds $|X|\le 2^{ps(X)}$ and $ψw_0(X)\le 2^{<ps(X)}$. 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.

math.GN↗
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.

math.CO↗
arXiv2025

On Proper Colorings of Functions

We investigate the infinite version of the $k$-switch problem of Greenwell and Lovász. Given infinite cardinals $κ$ and $λ$, for functions $x,y\in {}^λκ$ we say that they are totally different if $x(i)\ne y(i)$ for each $i\in λ$. A function $F:{}^λκ\longrightarrow κ $ is a proper coloring if $F(x)\ne F(y)$ whenever $x$ and $y$ are totally different elements of ${}^λκ $. We say that $F$ is weakly uniform iff there are pairwise totally different functions $\{r_α:α<κ\}\subset {}^λκ$ such that $F(r_α)=α$; $F$ is tight if there is no proper coloring $G:{}^λκ\longrightarrow κ$ such that there is exactly one $x\in {}^λκ$ with $G(x)\ne F(x)$. We show that given a proper coloring $F:{}^λκ\to κ$, the following statements are equivalent $F$ is weakly uniform, there is a $κ ^{+}$-complete ultrafilter $\mathscr{U}$ on $λ$ and there is a permutation $π\in Symm(κ)$ such that for each $x\in {}^λκ$ we have $$F(x)=π(α)\ \Longleftrightarrow \ \{i\in λ: x(i)=α\} \in \mathscr{U}.$$ We also show that there are tight proper colorings which cannot be obtained such a way.

math.CO↗
arXiv2024

Trivial coloring of Cartesian product of graphs

A coloring of a direct product of graphs is said to be {\em trivial} iff it is induced by some coloring of a factor of the product. A graph $G$ is trivially power colorable iff every coloring of a finite power of $G$ with $χ(G)$-many colors is trivial. Greenwell and Lovász proved that the finite complete graphs $K_n$ for $n\ge 3$ are trivially power colorable. Generalizing their result we define a much wider class of trivially power-colorable graphs: if $G$ is a finite, connected graph with $χ(G)\ge 3$ and every vertex of $G$ is in a clique of size $χ(G)$, then $G$ is trivially power-colorable. As an application of this result, we give a complete characterization of trivially power-colorable cographs. Finally, we give a structural description of the colorings of infinite powers of trivially power-colorable finite graphs.

math.CO↗
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

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↗
arXiv2021

Minimal vertex covers in infinite hypergraphs

In this paper a hypergraph will be identified with the family of its edges. A hypergraph $\mathcal E$ possesses property $C(k,ρ)$ iff $|\bigcap \mathcal E'|<ρ$ for each $\mathcal E'\in {[\mathcal E]}^{k}$. A vertex set $Y\subset \bigcup\mathcal E$ is a "vertex cover" of $\mathcal E$ iff $E\cap Y\ne \emptyset$ for each $E\in \mathcal E$. A vertex cover $Y$ is "minimal" iff no proper subset of $Y$ is vertex cover. If $A$ is a set and $S$ is a set of cardinals, write $$ {[A]}^{S}=\{B\subset A: |B|\in S\}. $$ If $λ$ and $ρ$ are cardinals, $S$ is a set of cardinals, $k\in ω$, then we write $$\mathbf M({λ},{S},{k},{μ})\to \mathbf{MinVC} $$ iff every hypergraph $\mathcal E\subset {[λ]}^{S}$ possessing property $C({k},{ρ})$ has a minimal vertex cover. If $S=\{κ\}$, then we simply write $\mathbf M({λ},κ,{k},{μ})\to \mathbf{MinVC}$ for $\mathbf M({λ},\{κ\},{k},{μ})\to \mathbf{MinVC}$ A set $S$ of cardinals is "nowhere stationary" iff $S\cap α$ is not stationary in $α$ for any ordinal $α$ with $cf(α)>ω$. Countable sets of cardinals, and sets of successor cardinals are nowhere stationary. In this paper we prove: (1) $\mathbf M({λ},{S},{2},{k})\to \mathbf{MinVC}$ for each nowhere stationary set $S$ of cardinals and $ω\le λ$, (2) $\mathbf M({λ},{κ} ,{2},{ρ})\to \mathbf{MinVC}$ provided $ρ<\beth_ω\le κ\le λ$, (3) $\mathbf M({λ},{ω},{r},{k})\to \mathbf{MinVC}$ provided $ω\le λ$ and $k,r\in ω$, (4) $\mathbf M({λ},{ω_1},{3},{k})\to \mathbf{MinVC}$ provided $ω_1\le λ$ and $k\in ω$.

math.LO↗