Remarks about van der Waerden ideal
FLAŠKOVÁ, J. Remarks about van der Waerden ideal. Polsko, Gdaňsk, 2013.
|Anglický název:||Remarks about van der Waerden ideal|
|Autoři:||RNDr. Jana Flašková Ph.D.|
|Abstrakt EN:||The sets which do not contain arbitrarily long arithmetic progressions form an ideal which we refer to as the van der Waerden ideal W. The ideal W is a tall Fsigma-ideal and it can be written as a countable union of strictly increasing Fsigma-ideals. We will observe that for every n>3 there exists a set A which does not contain any arithmetic progression of length n+1, but cannot be written as a finite union of sets with no arithmetic progressions of length n. We will, in particular, point out the set of square integers and consider its place in the structure of the ideal. The set does not contain any arithmetic progression of length 4, while it contains infinitely many arithmetic progressions of length 3. We will show that the question whether the set of squares can be written as finite union of sets with no arithmetic progressions of length 3 might be of relevance for the open problem about the existence of a perfect magic square of squares.|