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Ramsey numbers for partially ordered sets

Winter, Christian Malte ORCID iD icon 1
1 Institut für Algebra und Geometrie (IAG), Karlsruher Institut für Technologie (KIT)

Abstract:

The main objective of Ramsey theory is to investigate the largest monochromatic substructure guaranteed in any coloring of a given discrete host structure. Examples for such substructures are subgraphs hosted in a complete graph or arithmetic progressions in the natural numbers. In this thesis, we present quantitative Ramsey-type results in the setting of finite sets that are equipped with a partial order, so-called posets. A prominent example of a poset is the Boolean lattice $Q_n$, which consists of all subsets of $\{1,\dots,n\}$, ordered by inclusion. For posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the smallest $N$ such that no matter how the elements of $Q_N$ are colored in blue and red, there is either an induced subposet isomorphic to $P$ in which every element is colored blue, or an induced subposet isomorphic to $Q$ in which every element is colored red.

The central focus of this thesis is to investigate $R(P,Q_n)$, where $P$ is fixed and $n$ grows large. Our results contribute to an active area of discrete mathematics, which studies the existence of large homogeneous substructures in host structures with local constraints, introduced for graphs by Erd\H{o}s and Hajnal. ... mehr


Volltext §
DOI: 10.5445/IR/1000173294
Veröffentlicht am 20.08.2024
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Algebra und Geometrie (IAG)
Publikationstyp Hochschulschrift
Publikationsdatum 20.08.2024
Sprache Englisch
Identifikator KITopen-ID: 1000173294
Verlag Karlsruher Institut für Technologie (KIT)
Umfang xii, 172 S.
Art der Arbeit Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Algebra und Geometrie (IAG)
Prüfungsdatum 18.07.2024
Schlagwörter Ramsey theory, poset, Boolean lattice
Referent/Betreuer Axenovich, Maria
Martin, Ryan R.
Ueckerdt, Torsten
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