Aspects of Combinatorics and Combinatorial Number Theory

Aspects of Combinatorics and Combinatorial Number Theory Synopsis

The author discusses various Ramsey-type theorems in combinatorics and combinatorial number theory. While many of the main results are classic, recent progress and further open questions are described to motivate researchers. In classical theorems, wherever possible, proofs are kept different from those in Graham, Rothschild and Spencer's book. For instance, Johnson's proof has been given for Erdoes-Szekeres Theorem, mentioning references to other proofs. For Hilbert's theorem, after giving the dynamical proof following Furstenburg, a sketch of Hilbert's original proof is also given. The last part of the book describes many rather recent Ramsey-type results in combinatorics with application of topological ideas. It does not touch much of the graph theoretic part of the theory.;The first part of the book should appeal to anyone interested in the history, context and rudiments of the subject and requires only basic maturity in mathematical thinking. The later part and the remarks following each section should be of interest to professional mathematicians and useful for a person starting research in and around the subject. The advanced parts of the book require some mathematical training in algebra and topology.