Some Ramsey Properties of the Integers
Kevin Ventullo
Department of Applied Mathematics
Illinois Institute of Technology
Given a set of integers D, a D-diffsequence is a combinatorial object closely related to arithmetic progressions. It is defined to be any sequence a(1), a(2), ..., a(n) such that the consecutive differences (a(i) - a(i-1)) are in D for all 2 < i < n.
The degree of accessibility of a set D, denoted doa(D), is defined to be the greatest integer r such that every r-coloring of positive integers contains arbitrarily long monochromatic D-diffsequences. We will introduce and discuss some elementary properties and examples for these and other Ramsey-type concepts in a way that should be accessible to most undergraduates.
Our main result will be a new upper bound on doa(P+c), where P+c denotes the set of prime numbers translated by a constant c > 1.