3-accessibility of Fibonacci numbers

Importance: Medium ✭✭
Subject: Combinatorics
Recomm. for undergrads: no
Posted by: vjungic
on: August 24th, 2008

\begin{question} Is the set of Fibonacci numbers 3-accessible? \end{question}

A set $S$ is $r$-accessible if for any $r$-coloring of $\mathbb{N}$, $r\in \mathbb{N}$, there exist long monochromatic $S$-diffsequences, i.e., for any $k\in \mathbb{N}\backslash \{ 1\}$ there is a monochromatic sequence $\{ x_1,x_2,\ldots ,x_k\}$ such that $x_{i+1}-x_i \in S$ , for all $i\in \{ 1,2,\ldots ,k-1\}$.

The set of Fibonacci numbers $F$ is 2-accessible. \cite{LR1}

$F$ is not 6-accessible. \cite{AGJL}

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


[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.