Colouring homogeneous structures

Abstract

A relational structure is indivisible if for every partition of its set of elements into two parts there exists an embedding of the structure into one of the parts of the partition. A relational structure is homogeneous if every isomorphism of a finite induced substructure to a finite induced substructure extends to an automorphism. This article establishes a necessary and sufficient condition for Henson type, see [4], homogeneous structures to be indivisible.

Big Ramsey degrees using parameter spaces

“…By unrelated techniques, free amalgamation classes with the property that the big Ramsey degree of a vertex is equal to one were recently characterised by Sauer [Sau20]. Bounds on big Ramsey degrees of unrestricted structures with arities greater then 2 were announced in [BCH + 19] with a proof based on the vector (or product) form Milliken tree theorem [BCH + 20b, BCH + 20a].…”

Ramsey theory of homogeneous structures: current trends and open problems

“…Theorem 5.3 provides new classes of examples of indivisible Fraïssé structures, in particular for ordered structures such as the ordered Rado graph, while recovering results in [24], [42], and [27] and some of the results in [64]. Sauer's work in [64] provides the full picture on indivisibility for free amalgamation classes. Theorem 5.3 recovers certain cases of Sauer's results for FAP classes, while providing new SAP examples with indivisibility.…”

“…For homogeneous structures with free amalgamation with finitely many relations of any arity, the picture for indivisibility is now clear due to the recent breakthrough of Sauer. In [64], Sauer showed that a homogeneous free amalgamation structure with relations of finite arity is indivisible if and only if its age poset is linearly ordered, a property he called rank linear, culminating a line of work in [25], [26], [27], [61], and [28]. On the other hand, big Ramsey degrees of structures with relations of arity greater than two has only recently seen progress, beginning with [5], where Balko, Chodounský, Hubička, Konečný, and Vena found upper bounds for the big Ramsey degrees of the generic 3-hypergraph.…”

“…Theorem 5.4 provides new classes of examples of big Ramsey structures while recovering results in [10], [38], [43], and [44] and special cases of the results in [71]. Theorem 5.3 provides new classes of examples of indivisible Fraïssé structures, in particular for ordered structures such as the ordered Rado graph, while recovering results in [24], [42], and [27] and some of the results in [64]. Sauer's work in [64] provides the full picture on indivisibility for free amalgamation classes.…”