Monotone 4-term Arithmetic Progressions

Recomm. for undergrads: no
Posted by: vjungic
on: October 3rd, 2007

\begin{question} Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions? \end{conjecture}

It is not difficult to see that any permutation of positive integers contains a monotone 3-term arithmetic progression, i.e., that for any permutation $\pi :\mathbb{N}\to \mathbb{N}$ there is a 3-term arithmetic progression $a, a+d, a+2d$ such that $\pi (a)>\pi (a+d)>\pi (a+2d)$ or $\pi (a)<\pi (a+d)<\pi (a+2d)$.

In \cite{DEGS} an example of a permutation of $\mathbb{N}$ that does not contain a monotone 5-term arithmetic progression is given.


% Example: *[DEGS] J. A. Davis, R. C. Entringer, R. L. Graham, and G. J. Simmons, On permutations containing no long arithmetic progression, Acta Arithmetica XXXIV.1 (1977), 81-90.

[LR] Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Stud. Math. Libr. 24, AMS Providence, RI, 2004.

* indicates original appearance(s) of problem.