# Monotone 4-term Arithmetic Progressions

 Importance: Medium ✭✭
 Author(s): Davis, James A. Entringer, Roger C. Graham, Ronald L. Simmons, Gustavus J.
 Subject: Combinatorics
 Keywords: monotone arithmetic progression permutation
 Posted by: vjungic on: October 3rd, 2007
Question   Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions?

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 there is a 3-term arithmetic progression such that or .

In [DEGS] an example of a permutation of that does not contain a monotone 5-term arithmetic progression is given.

## Bibliography

*[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.