Titlebook: Mathematics of Ramsey Theory; Jaroslav Ne?et?il,Vojtěch R?dl Book 1990 Springer-Verlag Berlin Heidelberg 1990 Baum.Combinatorics.Komplexit

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Ordinal Types in Ramsey Theory and Well-Partial-Ordering TheoryThere is a gap between the infinite Ramsey’s theorem ω → (ω). and its finite version

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Partite Construction and Ramsey Space SystemsWe prove several Ramsey type theorems for parameter sets, affine and vector spaces by an amalgamation technique known as Partite Construction. This approach yields solution of several open problems and uniform treatment of several strongest results in the area. Particularly we prove Ramsey theorem for systems of spaces.

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Topics in Euclidean Ramsey TheoryMany questions in Ramsey Theory can be placed in the following context. We are given a set ., a family . of distinguished subsets of ., and a positive integer .. We would like to decide whether or not the following statement holds: For any partition of . = .. ∪…∪ .. into . classes, there is an . ∈ . and an index . such that . ? ...

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Shelah’s Proof of the Hales-Jewett Theoremplicity we include here only the proof of the one dimensional case of the theorem, which solves a problem of Graham by showing that the Hales-Jewett function is primitive recursive. The general cases will appear in the full paper of Shelah.