Ramsey Functions for Sequences with Restricted Gaps

摘要

For a set of positive integers D, a k-terrn D-diffsequence is a sequence of positive integers a_1 ＜ a_2 ＜ … ＜ a_k such that a_i - a_(i-1) ∈ D for all i ∈ {2, 3,..., k}. If k ∈ Z~+ and D ∈ Z~+, define Δ(D, k) to be the least positive integer n such that every 2-coloring of {1,2,. . . ,n} contains a monochromatic k-term D-diffsequence. Bounds and exact values for A(D,k) for certain choices of D are given, improving on previous results.