View source for Van der Waerden theorem

''on arithmetic progressions''

Given natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v0960501.png" />, there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v0960502.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v0960503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v0960504.png" /> is partitioned into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v0960505.png" /> sets, then at least one set contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v0960506.png" /> terms in [[Arithmetic progression|arithmetic progression]] [[#References|[a1]]]. Another, equivalent, non-finitary, formulation is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v0960507.png" /> be a finite partition of the natural numbers; then at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v0960508.png" /> contains arithmetic progressions of arbitrary length. The result was conjectured by A. Baudet. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v0960509.png" /> be a set of natural numbers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605010.png" /> be its upper [[Asymptotic density|asymptotic density]]. When discussing van der Waerden's theorem stated above, P. Erdös and P. Turán conjectured that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605012.png" /> contains arbitrary long arithmetic progressions; [[#References|[a2]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605013.png" /> be a maximally large subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605014.png" /> which contains no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605015.png" /> elements in arithmetic progression. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605016.png" /> be the number of elements in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605017.png" />. Then Szemerédi's theorem, [[#References|[a3]]], says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605018.png" />. This implies the Erdös–Turán conjecture. Another proof of Szemerédi's theorem was given by H. Furstenberg, based on ideas from [[Ergodic theory|ergodic theory]], [[#References|[a4]]].

For a personal historical account of the van der Waerden theorem see [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.L. van der Waerden,   "Beweis einer Baudetschen Vermutung"  ''Nieuw Arch. Wisk.'' , '''15'''  (1927)  pp. 212–216</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Erdös,   P. Turán,   "On some sequences of integers"  ''J. London Math. Soc.'' , '''11'''  (1936)  pp. 261–264</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Szemerédi,   "On sets of integers containing no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096050/v09605019.png" />-elements in arithmetic progression"  ''Acta Arithm.'' , '''27'''  (1975)  pp. 199–245</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Furstenberg,   "Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic progressions"  ''J. d'Anal. Math.'' , '''31'''  (1977)  pp. 204–256</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Furstenberg,   "Recurrence in ergodic theory and combinatorial number theory" , Princeton Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.Ya. Khinchin,   "Three pearls of number theory" , Graylock  (1952)  (Translated from Russian)</TD></TR></table>