Ramsey numbers for partially ordered sets
Christian Winter

TL;DR
This thesis investigates the asymptotic behavior of Ramsey numbers for posets, especially focusing on the Boolean lattice, providing tight bounds and exploring variations like non-induced subposets and color patterns.
Contribution
It offers asymptotically tight bounds for poset Ramsey numbers involving the Boolean lattice and identifies a dichotomy based on subposet structure, advancing understanding in poset Ramsey theory.
Findings
Established tight bounds on R(P,Q_n) for various posets P.
Identified a structural dichotomy affecting the asymptotic behavior of R(P,Q_n).
Improved bounds on R(Q_n,Q_n) for large n.
Abstract
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 , which consists of all subsets of , ordered by inclusion. For posets and , the poset Ramsey number is the smallest such that no matter how the elements of are colored in blue and red, there is either an induced subposet isomorphic to in which every element is colored blue, or an induced subposet isomorphic to in which every element is colored red. The central focus of this thesis is to investigate , where is fixed and 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…
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