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Notes on Number Theory II : On a theorem of van der Waerden
Canadian mathematical bulletin, Tome 3 (1960) no. 1, pp. 23-25

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A well known theorem of van der Waerden [1] states that given any two positive integers k and t, there exists a positive integer m such that in every distribution of the numbers 1,2, ..., m into k classes, at least one class contains an arithmetic progression of t + 1 terms. Other proofs and generalizations of this theorem have been given by Griinwald [2], Witt [3] and Lukomskaya [4]. The last mentioned proof appears in the booklet of Khinchin “Three pearls of number theory” in which van der Waerden's theorem plays the role of the first pearl. 
Notes on Number Theory II : On a theorem of van der Waerden. Canadian mathematical bulletin, Tome 3 (1960) no. 1, pp. 23-25. doi: 10.4153/CMB-1960-005-9
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[1] 1. van der Waerden, B. L., Beweis einer Baudetschen Vermutung, Nieuw Archief voor Wiskunde 15 (1921),212-216. Google Scholar

[2] 2. Radó, R., Note on combinatorial analysis, Proc. Lond. Math. Soc. 48 (1945), 122-160. Google Scholar

[3] 3. Witt, E., Ein kombinatorischer Satz der Elementargeometrie, Math. Nachrichten 6 (1952), 261-262. Google Scholar

[4] 4. Khinchin, A. Y., Three Pearls of Number Theory, (Rochester, 1952), 11-17. Google Scholar

[5] 5. Erdös, P. and Radó, R., Combinatorial theorems on classification of subsets of a given set, Proc. Lond. Math. Soc. 2 (1952), 417-439. Google Scholar

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