2-accessibility of primes

Importance: Medium ✭✭
Subject: Combinatorics
Recomm. for undergrads: no
Posted by: vjungic
on: July 9th, 2008

\begin{question} Is the set of prime numbers 2-accessible? \end{conjecture}

A set $S\subseteq \mathbb{N}$ 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 $i\in \{ 1,\ldots ,k-1\}$.

The set of primes $P$ is not 3-accessible. \cite{LR2}

Landman and Robertson proved \cite{LR1} that for any odd $t$, the set $t+P$ is 2-accessible.

It is known that a 2-coloring of any 33 consecutive positive integers yields a monochromatic 7-term $P$-diffsequence.


% Example:

[J] Jungi\'c, Veselin, {\it On a conjecture of Brown concerning accessible sets}, J. Combin. Theory Ser. A 110 (2005), \MRhref{MR2128973}

[KL] Abdollah Khodkar and Bruce M. Landman, {\it Recent progress in Ramsey theory on the integers}, Combinatorial number theory, 305--313, de Gruyter, Berlin, 2007. \MRhref{MR2337054}

[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. \MRhref{MR2354006}

*[LR2] Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Stud. Math. Libr. 24, AMS Providence, RI, 2004.

% %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.