We study the validity of a partition property known as weak indi-visibility for the integer and the rational Urysohn metric spaces. We also compare weak indivisiblity to another partition property, called age-indivisibility, and provide an example of a countable ultrahomogeneous metric space which is age-indivisible but not weakly indivisible.

We show that, for countable sofic groups, a Bernoulli action with infinite entropy base has infinite entropy with respect to every sofic approximation sequence. This builds on the work of Lewis Bowen in the case of finite entropy base and completes the computation of measure entropy for Bernoulli actions over countable sofic groups. One consequence is that such a Bernoulli action fails to have ...

Consider an arbitrary partition of the triples of all countable ordinals into two classes. We show that either for each finite ordinal m the first class of the partition contains all triples from a set of type ω +m, or for each finite ordinal n the second class of the partition contains all triples of an nelement set. That is, we prove that ω1 → (ω + m,n)3 for each pair of finite ordinals m and...

We answer a question of P. Cameron’s by giving examples of 2א0 many non-isomorphic acyclic orientations of the infinite random graph with a topological ordering that do not have the pigeonhole property. Our examples also embed each countable linear ordering. A graph is n-existentially closed or n-e.c. if for each n-subset S of vertices, and each subset T of S (possibly empty), there is a vertex...

A binary structure S has the pigeonhole property (P) if every finite partition of S induces a block isomorphic to S. We classify all countable tournaments with (P); the class of orders with (P) is completely classified.

Consider the skew product F : T2 → T2, F (x, y) = (f(x), y + τ(x)), where f : T1 → T1 is a piecewise C 1+α expanding map on a countable partition and τ : T1 → R is piecewise C 1. It is shown that if τ is not Lipschitzcohomologous to a piecewise constant function on the joint partition of f and τ , then F is mixing at a stretched-exponential rate.

Given a universal binary countable homogeneous structure U and n ∈ ω, there is a partition of the induced n-element substructures of U into finitely many classes so that for any partition C0, C1, . . . , Cm−1 of such a class Q into finitely many parts there is a number k ∈ m and a copy U∗ of U in U so that all of the induced n-element substructures of U∗ which are in Q are also in Ck. The parti...

For an ergodic p.m.p. action G y (X,μ) of a countable group G, we define the Rokhlin entropy hRok G (X,μ) to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov– Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Un...

A relation R is p-divisible if for any partition of its basis into p + 1 subsets, R is embedded into the union of p of them. We prove that any countable p-divisible relation embeds two copies of itself intersecting in at most p− 1 elements. A relation R is indivisible if for any partition of its set of vertices into two subsets V1 and V2, there exists i ∈ {1, 2} such that R is embedded into R(V...

In this paper we investigate edge partition problems of the countable triangle free homogeneous graph. As consequences of the main result, we obtain the following theorems. For every coloring of the edges of the countable triangle free homogeneous graph 0//with finitely many colors there exists a copy of q/in q/whose edges are colored with at most two of the colors. The countable triangle free ...