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By M. Lothaire

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To+{l — \)a) is contained in £„ t, contrary to the hypothesis. 2. 3. 2 the equivalence of the two statements is proved. 2. A Combinatorial Proof of van der Waerden's Theorem The present exposition follows that of Anderson (1976), who proves the following more general result, due to Grunwald (unpublished). 1. Let S be a finite subset of Nd. For each k-coloring ofNd there exists a positive integer a and a point v inNd such that the set aS + v is monochromatic. Moreover, the number a and the coordinates of the point v are bounded by a function that depends only on S and k {and not on the particular coloring used).

Iii) w2w, v2n are palindromes and u2n+ x = v2n+x. 2 Proof. The proofs are by induction. The initial step is always clear. Formula (i) follows from next (ii) follows from ^+i = ^«» = « » ^ = « ^ = « l l + i , t J I I + 1 = Sll+i = ii n+ finally for (iii), observe that for k > 0 If k is odd (resp. even) this implies uk = vk_xuk_x = vk (resp. uk = uk_xvk_x = uk). There exists an interesting definition of that is independent of the morphism /A. First let, for n>0yd2(n) be the number of l's in the binary expansion of n.

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 have Clearly c0 = 1, whence + l)2'. ck=f[{k-i 1 = 1 Consequently, M being the disjoint union of the ir(B), CardM= This completes the proof.

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