Rectangular square-bracket operation for successor of regular cardinals

Rinot, Assaf; Todorčević, Stevo

doi:10.4064/fm220-2-2

Cited by 11 publications

(7 citation statements)

References 8 publications

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“…Proof Recall that

κ^{+} ↛ {[κ^{+}; κ^{+}]}_{2}^{2}

holds whenever κ is a regular cardinal . Hence Theorem (1) and (2) implies that

\vec{χ} false(K_{κ^{+}} false) = κ^{+}

for any cardinal κ.…”

Section: Orientations Of Undirected Graphs With Large Chromatic Numbermentioning

confidence: 94%

Orientations of graphs with uncountable chromatic number

Soukup

2017

Journal of Graph Theory

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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 of size and chromatic number 1 has an orientation with uncountable dichromatic number" is independent of ZFC. We end the article with several open problems. K E Y W O R D Sacyclic, chromatic number, dichromatic number, digraph, girth, orientation, partition INTRODUCTIONThe chromatic number of an undirected graph , denoted by ( ), is the minimal number of independent sets needed to cover the vertex set of . A beautiful branch of graph theory deals with the 606

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“…Proof Recall that

κ^{+} ↛ {[κ^{+}; κ^{+}]}_{2}^{2}

holds whenever κ is a regular cardinal . Hence Theorem (1) and (2) implies that

\vec{χ} false(K_{κ^{+}} false) = κ^{+}

for any cardinal κ.…”

Section: Orientations Of Undirected Graphs With Large Chromatic Numbermentioning

confidence: 94%

Orientations of graphs with uncountable chromatic number

Soukup

2017

Journal of Graph Theory

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“…[ ; ] 2 holds for any cardinal that is the successor of an infinite regular cardinal; see [RT13] for an historical account and a uniform proof of the following: Fact 2.3 (Shelah, Moore). + [ + ; + ] 2 + holds for any infinite regular cardinal .…”

Section: The Foundations Of Walks On Ordinalsmentioning

confidence: 99%

Transformations of the transfinite plane

Rinot

Zhang

2021

Forum of Mathematics, Sigma

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We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every inaccessible cardinal $\kappa $ , if $\kappa $ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ implies that for every Abelian group $(G,+)$ of size $\kappa $ , there exists a map $f:G\rightarrow G$ such that for every $X\subseteq G$ of size $\kappa $ and every $g\in G$ , there exist $x\neq y$ in X such that $f(x+y)=g$ .

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“…Moore [20] developed ordinal-walks further and provided the missing κ + = ℵ 1 case. Rinot and Todorčević [29] present a unified proof of the rectangle version for all successors of regulars with a completely arithmetic oscillation function. Shelah, following Galvin [15], phrased the strong coloring relations Pr 1 (κ, µ, λ, χ) and Pr 0 (κ, µ, λ, χ) (and a few more!)…”

Section: A Brief History Of Strong Coloringsmentioning

confidence: 99%

Strong Colorings Over Partitions

Chen-Mertens¹,

Kojman²,

Steprāns³

2021

Bull. symb. log

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A strong coloring on a cardinal κ is a function f : [κ] 2 → κ such that for every A ⊆ κ of full size κ, every color γ < κ is attained by f [A] 2 . The symbolasserts the existence of a strong coloring on κ.We introduce the symbolWe prove that whenever κ [κ] 2 κ holds, also κ p [κ] 2 κ holds for an arbitrary finite partition p. Similarly, arbitrary finite p-s can be added to stronger symbols which hold in any model of ZFC. If κ θ = κ, then κ p [κ] 2 κ and stronger symbols, like Pr 1 (κ, κ, κ, χ) p or Pr 0 (κ, κ, κ, ℵ 0 ) p , also hold for an arbitrary partition p to θ parts.The symbols

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Rectangular square-bracket operation for successor of regular cardinals

Abstract: We give a uniform proof that λ + → [λ + ; λ + ] 2 λ + holds for every regular cardinal λ.

Cited by 11 publications

References 8 publications

Orientations of graphs with uncountable chromatic number

Orientations of graphs with uncountable chromatic number

Transformations of the transfinite plane

Strong Colorings Over Partitions

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