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, 2022 AbstractRamsey theory is a branch of combinatorics that asks questions of the form: How large must a set be so that if it is divided into subsets, at least one subset has a certain property? Results in Ramsey theory are difficult to prove and there remain many open problems.
Michael A. Henning, Jan H. van Vuuren
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Star-Critical Ramsey Numbers for Cycles Versus K4
Discussiones Mathematicae Graph Theory, 2021 Given three graphs G, H and K we write K → (G, H), if in any red/blue coloring of the edges of K there exists a red copy of G or a blue copy of H. The Ramsey number r(G, H) is defined as the smallest natural number n such that Kn → (G, H) and the star ...
Jayawardene Chula J. +2 more
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Ultraproducts and Related Constructions
Mathematics, 2022 In this work, we survey some research directions in which the ultraproduct construction and methods based on ultrafilters play significant roles. Rather different areas of mathematics have been considered: topics we are reviewing here include some ...
Gábor Sági
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A strict upper bound for size multipartite Ramsey numbers of paths versus stars
Indonesian Journal of Combinatorics, 2017 Let $P_n$ represent the path of size $n$. Let $K_{1,m-1}$ represent a star of size $m$ and be denoted by $S_{m}$. Given a two coloring of the edges of a complete graph $K_{j \times s}$ we say that $K_{j \times s}\rightarrow (P_n,S_{m+1})$ if there is a ...
Chula Jayawardene, Lilanthi Samarasekara
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Size multipartite Ramsey numbers for stripes versus small cycles
Electronic Journal of Graph Theory and Applications, 2016 For simple graphs $G_1$ and $G_2$, the size Ramsey multipartite number $m_j(G_1, G_2)$ is defined as the smallest natural number $s$ such that any arbitrary two coloring of the graph $K_{j \times s}$ using the colors red and blue, contains a red $G_1$ or
Chula Janak Jayawardene +3 more
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A Coloring Problem for Sturmian and Episturmian Words [PDF]
, 2013 We consider the following open question in the spirit of Ramsey theory: Given an aperiodic infinite word $w$, does there exist a finite coloring of its factors such that no factorization of $w$ is monochromatic? We show that such a coloring always exists
A. Glen +6 more
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Banach spaces and Ramsey Theory: some open problems [PDF]
, 2010 We discuss some open problems in the Geometry of Banach spaces having Ramsey-theoretic flavor. The problems are exposed together with well known results related to them.Comment: 17 pages, no figures; RACSAM, to ...
Dodos, Pandelis +2 more
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Anti-Ramsey theory on complete bipartite graphs
AKCE International Journal of Graphs and Combinatorics, 2020 We consider quadruples of positive integers with and such that every proper edge-coloring of the complete bipartite graph contains a rainbow subgraph. We show that every such quadruple with and satisfies this property and find an infinite sequence where ...
Stephan Cho +3 more
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, 2005 This paper presents an overview of the current state in research directions in the rainbow Ramsey theory. We list results, problems, and conjectures related to existence of rainbow arithmetic progressions in [n] and N. A general perspective on other rainbow Ramsey type problems is given.
Jungić, Veselin +2 more
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The Electronic Journal of Combinatorics, 2004 There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramsey-type theorems to various fields in mathematics are well documented in published books and monographs.
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