By M. Lothaire

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Classical Formulations Some forty years after he proved the theorem that bears his name, van der Waerden published an article (1965 and 1971) in which he describes the circumstances of the theorem's discovery. 1 mathematicians E. Artin, O. Schreier, and B. 1. //1^1 is partitioned into two classes, one of the classes contains arbitrarily long arithmetic progressions. The conjecture was extended by Artin to the case of a partition of N into k classes. 9Ek} of pairwise disjoint subsets of E whose union is E).

Setting JC = pay, there is a word sGA* such that Then x'=a/32sag and alph(a/? 2 s) = a l p h ( a ^ ) = a l p h ( x ) - {a} = a l p h ( x ' ) {a}. ' = a^2.? whencep''~ p,a — a'. The relations b = V,q~q' are proved in a symmetric manner. We now are ready to compute the number of elements in M = A*/~. Let 77: A* -> M be the canonical morphism and let, for 5 C ^4, Then ^4* is the disjoint union of the sets B,BCA. Since x~x' implies alph(x) = alph(x'), each B is a union of equivalence classes mod ~ , whence M is the disjoint union of the sets ir(B), B C A.

We distinguish two cases. ay, we have Case 1. \a/3\>\p\. a^ — pat, z — ty for some fin A + . Then x' = patfiy and alph(/>) = a l p h ( x ) - {«} = alph(x r ) — {a}. Thus by definition// = p and a' = a. Case 2. \afi\<\p\. Setting JC = pay, there is a word sGA* such that Then x'=a/32sag and alph(a/? 2 s) = a l p h ( a ^ ) = a l p h ( x ) - {a} = a l p h ( x ' ) {a}. ' = a^2.? whencep''~ p,a — a'. The relations b = V,q~q' are proved in a symmetric manner. We now are ready to compute the number of elements in M = A*/~.