
Re: Early (1930's) work on arithmetic progressions
Posted:
Sep 28, 2017 12:11 PM


On 09/28/2017 08:25 AM, Jack Campin wrote: >>>> I'm trying to understand some of the early work on arithmetic >>>> progressions. In the above paper, I don't understand how van der >>>> Waerden's theorem is used to prove (8) on page 236. It's true that >>>> one of the sets in the partition will contain arithmetic >>>> progressions of length k. But, in the partition of the integers >>>> from 1 to r_k * n_0, an A_i will have lots of members which are >>>> not in the a_i maximal sequence. So I don't see how an A_i with an >>>> arithmetical progression of length k establishes a contradiction. > >> http://dml.cz/bitstream/handle/10338.dmlcz/122006/CasPestMatFys_06719384_3.pdf >> >> should be openable by all. PDF, not being particularily friendly, is >> a deterent to considering your question. > > Nothing wrong with it. Opens just fine in either Apple Preview (which > ignores the certification stuff) or Adobe Reader (which on my system > can be asked to ignore it). > > I've no idea about the argument, though. > >  > e m a i l : j a c k @ c a m p i n . m e . u k > Jack Campin, 11 Third Street, Newtongrange, Midlothian EH22 4PU, Scotland > mobile 07895 860 060 <http://www.campin.me.uk> Twitter: JackCampin >
Some might be interested in later developments on sequences of integers in [1, n] that avoid arithmetic progressions of length k = 3 ( other cases are k =4, k = 5 and so on).
Endre Szemerédi has a presentation in the 1974 Proceedings of the International Congress of Mathematicians.
It refers to a paper of Behrend from 1938, and another from 1946:
http://www.mathunion.org/ICM/ICM1974.2/Main/icm1974.2.0503.0508.ocr.pdf
David Bernier

