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
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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 -