A short proof of van der Waerden’s theorem on arithmetic progressions

A short proof is given for the classical theorem of van der Waerden which asserts that for any partition of the integers into a finite number of classes, some class contains arbitrarily long arithmetic progressions. Let [a, b] denote the set of integersxwith a_x_b. We call (x1, ** ,x (x', * x*m) E [0, r]m I-equivalent if they agree up through their last occurrences of i. For any 1, m> 1, consider the statement For any r, there exists N(I, m, r) so that for any function S(, m) C: [1, N(i, m, r)]-.[1, r], there exist positive a, d,, * *, dm such that C(a+ Im 1 xidi) is constant on each 1-equivalence class of [0, i]m. FACT 1. S(i, m)for some m >?I=S(l,m+1). PROOF. For a fixed r, let M=N(I, m, r), M'=N(l, 1, rM) and suppose C: [1, MM']-[1, r] is given. Define C': [1, M']-*.[1, rM] so that C'(k)= C'(k') iff C(kM-j)=C(k'M-j) for all 0_ j l=>S(I+1, 1). PROOF. For a fixed r, let C: [1, 2N(l, r, r)]1,r] be given. Then there exist a, dl, * , dr such that for xi E [0, 1], a+ jr 1 xidj<N(i, r, r) and C(a+ 22=1 xidi) is constant on I-equivalence classes. By the box principle there exist u<v in [0, r] such that C(a + E Is) =C (a+2id)Received by the editors February 1, 1973. A MS (MOS) subject classifcations (1970). Primary 05A99; Secondary lOL99.