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Journal of Combinatorial Theory, Series A, 1974 AbstractA subset S of a group G is said to be a sum-free set if S ∩ (S + S) = ⊘. Such a set is maximal if for every sum-free set T ⊆ G, we have |T| ⩽ |S|. Here, we generalize this concept, defining a sum-free set S to be locally maximal if for every sum free set T such that S ⊆ T ⊆ G, we have S = T.
Street, Anne Penfold +1 more
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The Ramsey theory of the universal homogeneous triangle-free graph [PDF]
Journal of Mathematical Logic, 2017 The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math. 38(1) (1971) 69–83] and denoted [Formula: see text], is the triangle-free analogue of the Rado graph. While the Ramsey theory
Natasha Dobrinen
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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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Ramsey theory without pigeonhole principle and the adversarial Ramsey principle [PDF]
Transactions of the American Mathematical Society, 2018 We develop a general framework for infinite-dimensional Ramsey theory with and without pigeonhole principle, inspired by Gowers’ Ramsey-type theorem for block sequences in Banach spaces and by its exact version proved by Rosendal.
N. D. Rancourt
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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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European Journal of Combinatorics, 1997 AbstractLet G be a countable graph which has infinite chromatic number. Ifγis a coloring of [G]2with two colors, is there then a subsetH⊆Gsuch thatγis constant on [H]2andG|H,the graph induced by G onH,has infinite chromatic number? As edges and non-edges can be colored with different colors this will be the case iff G contains an infinite clique.
Norbert Sauer +2 more
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Discrete Mathematics, 1987 AbstractLet [n] denote the n-set {1, 2,..., n}, let k, l ⩾ 1 be integers. Define fl(n, k) as the minimum number f such that for every family F ⊆ 2[n] with ∣F∣>f, for every k-coloring of [n], there exists a chain A1⫋···⫋Al+1 in F in which the set of added elements, Al+1−A1, is monochromatic.We survey the known results for l = 1.
Zoltán Füredi +3 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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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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