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Soft cover. Condition: New. 1st Edition. 8vo (22.5 cm), 148 pp. Publisher's laminated wrappers. Mathematical Studies: Monograph Series, vol. 11. Edited by Michael Zarichnyi. A research monograph in combinatorics, graph theory, Ramsey theory, and geometric group theory by Ihor Protasov (Kyiv National University) and Taras Banakh…(Lviv National University), developing a unified framework for studying colorings and large-scale structures in graphs and groups. The authors consider several classes of hypergraph and graph colorings--including surjective, injective, nonconstant, and kaleidoscopical (bijective) colorings--and use this framework to connect classical results such as the theorems of Van der Waerden and Gallai with their own results, notably the "Grasshopper Theorem" concerning quasicycles in finite graphs. The thirteen sections progress from balanced partitions of finite and infinite graphs to quasicycles, quasirays, and quasihamiltonian graphs; chromatic numbers; kaleidoscopical graphs and groups; and the introduction of ball structures, an abstract framework encompassing the notion of a ball in a metric space, graph, or group. The authors study morphisms of ball structures and apply the theory to questions concerning generating sets of groups. Among the more distinctive topics are the "botanic" characterization of trees admitting quasirays through the notion of a trunk; kaleidoscopical semigroups KS(X); and contrasting examples involving the group A∞ of even permutations of ℕ and SO(3), illustrating markedly different behaviour of finite kaleidoscopical subsets. Each section concludes with exercises and open problems. A specialized and original contribution to infinite combinatorics and large-scale algebraic structures, of particular interest to researchers working in Ramsey theory, graph theory, geometric group theory, and asymptotic topology.
