Ramsey properties and extending partial automorphisms for classes of finite structures

Abstract

We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Nešetřil-Rödl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results solve several conjectures and provide an easy Ramseyness test for many classes of structures.

30 pages, 9 figures; corrections and presentation improvements suggested by the referee. Functions in structures are now set-valued

References in corpus (3)

• Ramsey expansions of metrically homogeneous graphs
• A Ramsey Class for Steiner Systems
• Universal graphs with a forbidden subgraph: Block path solidity

Cited by in corpus (9)

• All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)
• Ramsey expansions of metrically homogeneous graphs
• Semigroup-valued metric spaces
• EPPA numbers of graphs
• Extending partial automorphisms of $n$-partite tournaments
• Structural Ramsey Theory and the Extension Property for Partial Automorphisms
• Supersaturation Problem for the Bowtie
• Extension property for partial automorphisms of the $n$-partite and semigeneric tournaments
• Twenty years of Nešetřil's classification programme of Ramsey classes