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Primitive recursive bounds for van der Waerden numbers. (English) Zbl 0649.05010
The author solves one of the most outstanding and notorious problems of combinatorics by exhibiting primitive recursive bounds on the van der Waerden numbers. All earlier proofs gave Ackermann-like bounds and the question on the existence of primitive recursive bounds was posed by Solovay, and widely popularized by Erdős, Graham, and others. “Only time will tell, and we may not be here to listen, what the truth is” wrote Graham on the problem [R. L. Graham; Rudiments of Ramsey theory (1981; Zbl 0458.05043)]. We are here and eagerly listening to the amazingly simple solution presented here.
If n is the length of the arithmetical progression to be found, c is the number of colors, the orthodox proofs for the case (n,c) used the case (n-1,d) where d is a very large number. This double induction is responsible for the Ackermann-like growth. The proof here presented uses only simple induction, i.e. the case (n,c) is deduced from the case (n- 1,c). A relatively simple Ramsey type lemma is used, for which the application of the pidgeonhole principle gives superexponential but primitive recursive bounds. Then, the Hales-Jewett theorem is deduced, along with the Graham-Rothschild higher dimensional generalization and the affine Ramsey theorem on monocolored subspaces. This important problem on van der Waerden numbers though resolved the real behaviour of the growth of this function still remains a mystery.
Reviewer: P.Komjáth

05A99 Enumerative combinatorics
05C55 Generalized Ramsey theory
11B25 Arithmetic progressions
15A03 Vector spaces, linear dependence, rank, lineability
03D20 Recursive functions and relations, subrecursive hierarchies
Full Text: DOI
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