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Integer Sets Containing No Arithmetic Progressions
Author(s) -
HeathBrown D. R.
Publication year - 1987
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-35.3.385
Subject(s) - citation , integer (computer science) , arithmetic , computer science , mathematics , information retrieval , combinatorics , library science , programming language
lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of length k. This is the famous theorem of van der Waerden [10], dating from 1927. The proof of this uses multiple nested inductions, which result in extremely weak bounds for N{h,k). We shall define Bk to be the collection of all sets si £= N for which sf contains no arithmetic progression of length k. We then set