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Quantitative versions of combinatorial partition theorems. (Quantitative Versionen von kombinatorischen Partitionssätzen.) (German) Zbl 0737.05012
There are many “Ramsey theorems”, the most famous of which are Schur’s theorem (see Über die Kongruenz \(x^ m+y^ m\equiv z^ m(\mod p)\), Jahresber. Dtsch. Math.-Ver. 25, 114-116 (1916)), van der Waerden’s theorem (see B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15, 212-216 (1927)), and Ramsey’s theorem (see F. P. Ramsey, On a problem in formal logic, Proc. London Math. Soc. (2) 30, 264-285 (1930)). These theorems all all deal with partitions of the infinite set of natural numbers. In the present paper the authors are concerned with theorems of a similar kind, except that they now deal with partitions of a finite set of natural numbers \(\{1,2,\ldots,N\}\).
05A17 Combinatorial aspects of partitions of integers
05C55 Generalized Ramsey theory
11P81 Elementary theory of partitions