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Van der Waerden's theorem on arithmetic progressions

Student-Run Research Seminar

Speaker: Momar Dieng, UC Davis
Location: 693 Kerr
Start time: Wed, Apr 27 2005, 12:10PM

Van Der Waerden's theorem states that if and are two arbitrary natural numbers, there exists a natural number such that if any arbitrary segment of the sequence of natural numbers of length is divided in ANY manner into classes (some of which may be empty), then an arithmetic progression of length appears in at least one of these classes. I will give an elementary proof of it, originally due to M. A. Lukomskaya.

Van Der Waerden's theorem is a corollary of Szemerédi's theorem, which states that every sequence of integers that has positive upper Banach density (to be defined) contains arbitrarily long arithmetic progressions. Szemerédi's theorem was conjectured by Erdos and Turán (1936). Roth (1953) proved the case = 3, which was mentioned in his Fields Medal citation. Szemerédi (1969) proved the case = 4, and the general theorem in 1975 as a consequence of the so-called Szemerédi Regularity Lemma, for which he collected a N(K,L)N(K,L)K$ = 4 and this was mentioned in his Fields Medal citation. If time permits we will go into this.