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Topic: Early (1930's) work on arithmetic progressions
Replies: 10   Last Post: Oct 1, 2017 11:40 AM

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David Bernier

Posts: 3,884
Registered: 12/13/04
Re: Early (1930's) work on arithmetic progressions
Posted: Sep 28, 2017 12:11 PM
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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_067-1938-4_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




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