Logic Seminar

Please note that the time is PST

Given a finite relational language LL and a (possibly infinite) set FF of finite irreducible LL-structures, the class Forb(F)Forb⁡(F) describes those finite LL-structures which do not embed any member of FF. Classes of the form Forb(F)Forb⁡(F) exactly describe those classes of finite LL-structures with free amalgamation. In recent joint work with Balko, Chodounsky, Dobrinen, Hubicka, Konecny, and Vena, we exactly characterize big Ramsey degrees for those classes Forb(F)Forb⁡(F) where the forbidden set FF is finite. This characterization proceeds by defining tree-like objects called diagonal diaries, then showing that the big Ramsey degree of any AA in Forb(F)Forb⁡(F) is exactly the number of diagonal diaries which code the structure AA. After giving a brief description of these objects, the talk will then consider those infinite diagonal diaries which code the Fraisse limit of Forb(F)Forb⁡(F). In upcoming joint work with Dobrinen, we prove a Galvin-Prikry theorem for any such infinite diagonal diary, giving new examples of objects satisfying the Galvin-Prikry theorem which dramatically fail to satisfy Todorcevic's Ramsey space axioms A1 through A4.