TY - JOUR

T1 - Avoiding monochromatic sequences with special gaps

AU - Landman, Bruce M.

AU - Robertson, Aaron

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2007

Y1 - 2007

N2 - For S ⊆ ℤ+ and k and r fixed positive integers, denote by f(S, k; r) the least positive integer n (if it exists) such that within every r-coloring of {1,2,. .., n} there must be a monochromatic sequence {x 1, x2,....xk} with xi - x i-1 ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S, k;r) for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if 5 is an odd translate of the set of primes and r = 2.

AB - For S ⊆ ℤ+ and k and r fixed positive integers, denote by f(S, k; r) the least positive integer n (if it exists) such that within every r-coloring of {1,2,. .., n} there must be a monochromatic sequence {x 1, x2,....xk} with xi - x i-1 ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S, k;r) for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if 5 is an odd translate of the set of primes and r = 2.

KW - Arithmetic progressions

KW - Primes in progression

KW - Ramsey theory

UR - http://www.scopus.com/inward/record.url?scp=49449102333&partnerID=8YFLogxK

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U2 - 10.1137/S0895480103422196

DO - 10.1137/S0895480103422196

M3 - Article

AN - SCOPUS:49449102333

SN - 0895-4801

VL - 21

SP - 794

EP - 801

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 3

ER -