Ramsey theory and partially ordered sets

Abstract

Over the past 15 years, Ramsey theoretic techniques and concepts have been applied with great success to partially ordered sets. In the last year alone, four new applications of Ramsey theory to posets have produced solutions to some challenging combinatorial problems. First, Kierstead and Trotter showed that dimension for interval orders can be characterized by a single ramsey trail by proving that interval orders of sufficiently large dimension contain all small interval orders as subposets. Second, Winkler and Trotter introduced a notion of Ramsey theory for probability spaces and used the resulting theroy to show that interval orders can have fractional dimension arbitrarily close to 4. Third, Felsner, Fishburn and Trotter developed an extension of the product Ramsey theorem to show that there exists a finite 3-dimensional poset which is not a sphere order. Fourth, Agnarsson, Felsner and Trotter combined Ramsey theoretic techniques with other combinatorial tools to determine an asymtotic formula for the maximum number of edges in a graph whose incidence poset has dimension at most 4. In this paper, we outline how these applications were developed. Full details will appear in individual journal articles. This article also includes a brief sketch of how the applications of Ramsey theoretic techniques to posets have evolved.