# 2-accessibility of primes

 Importance: Medium ✭✭
 Author(s): Landman, Bruce M. Robertson, Aaron
 Subject: Combinatorics
 Keywords: monochromatic diffsequences primes
 Posted by: vjungic on: July 9th, 2008
Question   Is the set of prime numbers 2-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 primes is not 3-accessible. [LR2]

Landman and Robertson proved [LR1] that for any odd , the set is 2-accessible.

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

## Bibliography

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

[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. MathSciNet

[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. MathSciNet

*[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.