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Contains a complete presentation of some highly sophisticated proofs following new methods of pushing and pulling
Includes several diametric theorems for sequence spaces, bounds for list codes, and even the concepts of higher level and dimension constrained extremal problems, and splitting antichains
Examines number-theoretical and combinatorial extremal problems in parallel, which makes the connections more transparent
The main focus of these lectures is basis extremal problems and inequalities – two sides of the same coin. Additionally they prepare well for approaches and methods useful and applicable in a broader mathematical context.
Highlights of the book include a solution to the famous 4m-conjecture of Erdös/Ko/Rado 1938, one of the oldest problems in combinatorial extremal theory, an answer to a question of Erdös (1962) in combinatorial number theory "What is the maximal cardinality of a set of numbers smaller than n with no k+1 of its members pair wise relatively prime?", and the discovery that the AD-inequality implies more general and sharper number theoretical inequalities than for instance Behrend's inequality.
Several concepts and problems in the book arise in response to or by rephrasing questions from information theory, computer science, statistical physics. The interdisciplinary character creates an atmosphere rich of incentives for new discoveries and lends Ars Combinatoria a special status in mathematics.
At the end of each chapter, problems are presented in addition to exercises and sometimes conjectures that can open a reader’s eyes to new interconnections.
Content Level »Graduate
Keywords »AZ-identities - LYM-inequality - Maxima - Number theory - Prime - antichain splitting - combinatorics - diametric problems - intersection problems