Ramsey theory on relational structures has been investigated ever since Ramsey proved his seminal theorem. While a multitude of classes of finite structures have been shown to possess the Ramsey property, such as finite linear orders and finite ordered graphs, analogues for infinite structures have proven more elusive: Initiated by Sierpi ́nski in the 1930’s, it was not until 1979 that the Ramsey theory of the rationals as a linearly ordered structure was completely understood; the Ramsey theory of the Rado graph was only completed in 2006. The main reason for this is that methods for Ramsey theory on finite structures are in general not sufficient for discovering Ramsey properties of their homogeneous (infinite) counterparts, i.e., Fra ̈ıss ́e limits. Furthermore, while methods used for the rationals and Rado graph aided in a few other results, those methods were not sufficient for infinite structures with certain forbidden substructures, such as triangle-free graphs, hindering further progress in the area.
We will discuss the main reasons why this is so, and how the set-theoretic method of forcing opened new pathways in this arena. We
will indicate the structural properties involved in the Ramsey theory of various homogeneous structures such as the rationals, the Rado
graph, the homogeneous triangle-free graph and binary relational free-amalgamation classes more generally, the generic partial order, and others, indicating similarities and differences between their Ramsey-theoretic characterizations. Ongoing work as well as cognate areas, including infinite-dimensional Ramsey theory of homogeneous structures and partition theory of uncountable structures, will also be discussed. This talk will include works of various author combinations from among Balko, Barbosa, Chodounsk ́y, Coulson, El-Zahar, Erd ̋os, Hajnal, Hubiˇcka, Komj ́ath, Koneˇcn ́y, Laflamme, Larson, Maˇsulovi ́c, Neˇsetˇril, Nguyen Van Th ́e, Patel, P ́osa, R ̈odl, Sauer, Vena, Zucker and the speaker.