Ramsey Degrees Research Articles

SEMILATTICES AND THE RAMSEY PROPERTY

AbstractWe consider${\cal S}$, the class of finite semilattices;${\cal T}$, the class of finite treeable semilattices; and${{\cal T}_m}$, the subclass of${\cal T}$which contains trees with branching bounded bym. We prove that${\cal E}{\cal S}$, the class of finite lattices with linear extensions, is a Ramsey class. We calculate Ramsey degrees for structures in${\cal S}$,${\cal T}$, and${{\cal T}_m}$. In addition to this we give a topological interpretation of our results and we apply our result to canonization of linear orderings on finite semilattices. In particular, we give an example of a Fraïssé class${\cal K}$which is not a Hrushovski class, and for which the automorphism group of the Fraïssé limit of${\cal K}$is not extremely amenable (with the infinite universal minimal flow) but is uniquely ergodic.

Unary functions

We consider F the class of finite unary functions, B the class of finite bijections and Fk, k>1, the class of finite k−1 functions. We calculate Ramsey degrees for structures in F and Fk, and we show that B is a Ramsey class. We prove Ramsey property for the class OF which contains structures of the form (A,f,≤) where (A,f)∈F and ≤is a linear ordering on the set A. We also consider a generalization MnF, n>1, of the class F which contains finite structures of the form (A,f1,...,fn) where each fi is a unary function on the set A. Finally we give a topological interpretation of our results by expanding the list of extremely amenable groups and by calculating various universal minimal flows.

Directed graphs and boron trees

Let L1 and L2 be two disjoint relational signatures. Let K1 and K2 be Ramsey classes of rigid relational structures in L1 and L2 respectively. Let K1⁎K2 be the class of structures in L1∪L2 whose reducts to L1 and L2 belong to K1 and K2 respectively. We give a condition on K1 and K2 which implies that K1⁎K2 is a Ramsey class. This is an extension of a result of M. Bodirsky.In the second part of this paper we consider classes OS(2), OS(3), OB and OH which are obtained by expanding the class of finite dense local orders, the class of finite circular directed graphs, the class of finite boron tree structures, and the class of rooted trees respectively with linear orderings. We calculate Ramsey degrees for objects in these classes.

Topological dynamics of unordered Ramsey structures

We investigate the connections between Ramsey properties of Fraisse classes K and the universal minimal flow M(GK) of the automorphism group GK of their Fraisse limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class K has finite Ramsey degree for embeddings, then this degree equals the size of M(GK). We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if K is a relational Ramsey class and GK is amenable, then M(GK) admits a unique invariant Borel probability measure that is concentrated on a unique generic orbit.

The shift graph and the Ramsey degree of ${[\mathbb{N}]}^{\omega}$

We address several questions related to the Ramsey degree of the class of infinite structures isomorphic to the positive integers with the usual ordering. In particular we study the strength of stating that this class has finite Ramsey degree in the absence of the axiom of choice. We show that if the shift graph has finite chromatic number then this class has infinite Ramsey degree.

Ramsey degrees of boron tree structures

We investigate the combinatorial properties of the Fraisse class of structures induced by the leaf sets of boron trees -- graph-theoretic binary trees without an assigned root -- and compute their Ramsey degrees. The Ramsey degree of a boron tree structure is shown to equal the number of its possible orientations, which are herein defined to depend on the embedding of the said structure. One direction of this computation involves an asymmetric version of the Graham-Rothschild theorem. By expanding these structures to oriented ones, we arrive at a Fraisse class which possesses the Ramsey property. Consequently, the automorphism group of its Fraisse limit is extremely amenable, i.e., it possesses a very strong fixed point property. Furthermore, we construct the universal minimal flow of the automorphism group of this limit.

Ramsey degrees of finite ultrametric spaces, ultrametric Urysohn spaces and dynamics of their isometry groups

We study Ramsey-theoretic properties of several natural classes of finite ultrametric spaces, describe the corresponding Urysohn spaces, and compute a dynamical invariant attached to their isometry groups.

Big Ramsey Degrees and Divisibility in Classes of Ultrametric Spaces

AbstractGiven a countable set S of positive reals, we study finite-dimensional Ramsey-theoretic properties of the countable ultrametric Urysohn space QS with distances in S.

Ramsey degrees of bipartite graphs: A primitive recursive proof

In this paper, we give a primitive recursive proof for a theorem that provides a measure of the extent to which a finite bipartite graph is a Ramsey object in the class of finite bipartite graphs.

Symmetry and the Ramsey Degrees of Finite Relational Structures

In this paper, we introduce a measure of the extent to which a finite combinatorial structure is a Ramsey object in the class of objects with a similar structure. We show for classes of finite relational structures, including graphs, binary posets, and bipartite graphs, how this measure depends on the symmetries of the structure.

Symmetry and the Ramsey degree of posets

In this paper we introduce a measure of the extent to which a given finite poset deviates from being a Ramsey object in the class of finite posets. We show how this measure depends on the symmetry properties of a poset.