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Ramsey properties for integer sequences with restricted gaps

Bruce M. Landman, Aaron Robertson and Quinn Robertson

Vol. 12 (2023), No. 3, 181–195
Abstract

For D + , a sequence x1 < x2 < < xk is called a D-diffsequence if xi xi1 D for all i {2,,k}. Given a positive integer r, we say that D + is r-accessible if every r-coloring of + admits arbitrarily long monochromatic D-diffsequences. If every r-coloring of + admits arbitrarily long monochromatic arithmetic progressions xx + d, x + 2d, , x + (k 1)d, with d D, we say that D is r-large. We prove some new results on accessibility. We also define a property that is stronger than accessibility but weaker than largeness, and study the connections among these various properties. The paper also includes bounds, as well as tables of exact values, for the associated Ramsey functions for some specific choices of D.

Keywords
van der Waerden, diffsequence, Ramsey theory
Mathematical Subject Classification
Primary: 05D10, 11B25
Secondary: 11B39
Milestones
Received: 21 January 2023
Accepted: 11 June 2023
Published: 23 September 2023
Authors
Bruce M. Landman
Department of Mathematics
University of Georgia
Athens, GA
United States
Aaron Robertson
Department of Mathematics
Colgate University
Hamilton, NY
United States
Quinn Robertson
Technology Services Department
SUNY Morrisville
Morrisville, NY
United States