Aspects of Combinatorics and Combinatorial Number Theory
Aspects of Combinatorics and Combinatorial Number Theory discusses various Ramsey-type theorems in combinatorics and combinatorial number theory. While many of the main results are classic, the book describes recent progress and considers unsolved questions in the field. For classical theorems, whenever possible, the author presents different proofs than those offered in Graham, Rothschild, and Spencer's book. For instance, Johnson's proof has been given for Erdoes-Szekeres Theorem, and in establishing that proof, the author makes reference to the other proofs. The first part of the book is primarily concerned with the history, context, and rudiments of the subject, and it requires only a basic maturity in mathematical thinking. The later parts and the remarks following each section describe many rather recent Ramsey-type results in combinatorics with application of topological ideas. These parts require some training in algebra and topology.
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2-subsets 4.1 of Chapter arithmetic progression assume belong coefficient colouring of Z+ compact semigroup congruent conjecture consider contains an arithmetic corresponding defined denote the number distinct elements equation equivalence relation Erdos Euclidean EXERCISE 2.1 finite colouring finite set fliptop Furstenberg and Katznelson graph G graphical representation Hales-Jewett theorem hence Hindman's theorem idempotent implies induction left ideal Lemma length linear matrix minimal left ideal modulo monochromatic combinatorial line monochromatic set monochromatic solution monochromatic subset natural density non-empty subsets non-zero notation NU(T NU(V number of partitions observe obtained pairwise disjoint partition function pigeonhole principle polynomial posets positive integer prime proof of Theorem prove r-colouring Rado Ramsey theory Ramsey's theorem real numbers REMARK result Rodl satisfying Schur's theorem Section semigroup sequence Shelah Shelah line Sidon set statement Szemeredi Theorem 3.1 theorem Theorem topological triangle vector vertices Waerden's theorem words Z+)d zero