# 3-accessibility of Fibonacci numbers

 Importance: Medium ✭✭
 Author(s): Landman, Bruce M. Robertson, Aaron
 Subject: Combinatorics
 Keywords: Fibonacci numbers monochromatic diffsequences
 Posted by: vjungic on: August 24th, 2008
Question   Is the set of Fibonacci numbers 3-accessible?

A set is -accessible if for any -coloring of , , there exist long monochromatic -diffsequences, i.e., for any there is a monochromatic sequence such that , for all .

The set of Fibonacci numbers is 2-accessible. [LR1]

is not 6-accessible. [AGJL]

It is known that a 3-coloring of any 27 consecutive positive integers yields a monochromatic 4-term -diffsequence.

## Bibliography

[AGJL] Hayri Ardal, David Gunderson, Veselin Jungi\'c, and Bruce Landman, {\it On Accessibility of the Set of Fibonacci Numbers}, In Preparation

*[LR1] Bruce M. Landman and Aaron Robertson, {\it Avoiding Monochromatic Sequences With special Gaps}, SIAM J. Discrete Math Vol. 21 (2007), no. 3, 794--801.

[LR2] 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.