On monochromatic arithmetic progressions having odd step. (English) Zbl 0965.05093

Jungic, Veselin

Exp. Math. 9, No. 3, 467-471 (2000).

The van der Waerden theorem states that in any two-coloring of the integers there are arbitrary long arithmetic progressions; but the even/odd-coloring shows that we can easily avoid all arithmetic progressions with odd step. The author studies the properties of other two-colorings with respect to the existence of a monochromatic arithmetic progression with odd step, and poses the question: is there a \(t\in]0,1[\) such that, if in a two-coloring of the integers in each congruence class modulo \(2p\) the upper density of color 0 is at least \(t\), there is an arithmetic progression of color 0 with odd step and length \(2p\)? The author describes some experimental work on this conjecture for the case \(p=2\) with \(t={1\over 2}\).

Reviewer: Peter Braß (Berlin)

MSC:

05D10	Ramsey theory
11B25	Arithmetic progressions

Keywords:

monochromatic arithmetic progressions; van der Waerden theorem; colorings of the integers

References:

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