Avoiding arithmetic progressions (mod m) and arithmetic progressions
Abstract
A family F of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer f = f(r)(F,k) such that for every r-coloring of {1,2,...,f} there is a monochromatic k-term member of F. For fixed integers 1 ≤ a < m, a k-term a(mod m)-sequence is an increasing sequence of positive integers {x1,... ,xk} such that xi-xi ≡ a(mod m) for i = 2,... ,k. An arithmetic progression modulo m is a sequence that is an a(mod m)-sequence for some a ∈ {1,2,... , m - 1}. A d-a.p. is an arithmetic progression where the difference between successive terms is d. It is known that if 1 ≤ a < m are fixed, then the family consisting of all a(mod m)-sequences and all m-a.p.'s has the 2-Ramsey property, does not have the 4-Ramsey property, and has the 3-Ramsey property only when a = m/2. This paper extends these results to much larger families. In particular, we show that if D contains at most a finite number of multiples of m, then the family of sequences that are either arithmetic progressions modulo m, or are d-a.p.'s for some d ∈ D, does not have the 4-Ramsey property, and has the 3-Ramsey property if and only if m is even and D contains some multiple of m.
