Let $G$ be an infinitely countable amenable group. We say that a symbolic system $(X,G)$ has specification with margin $M$ if for any two subsets $F_1,F_2$ of $G$, such that $MF_1\cap MF_2=\varnothing$, any $X$-admissible pattern over $F_1$ and any $X$-admissible pattern over $F_2$ coexist in some element of $X$. We show that if a symbolic system with specification $(X,G)$ supports at least one free measure, then it is universal in the sense that for every free ergodic measure-theoretic system $(Y,\nu,G)$ of entropy smaller than (topological) entropy of $(X,G)$, there exists a measure $\mu$ supported by $X$, which is isomorphic to $\nu$. An important element of the proof of the theorem is the use of markers, the construction of which was presented in the previous talk by Tomasz Downarowicz.

Suppose you live on a $\mathbb Z\times \mathbb Z$ lattice. Imagine that you want to precisely (up to an integer vector of coordinates) mark a landing spot for a desant squad expected at night. You are allowed to use a finite configuration of lights positioned at the elements of $\mathbb Z\times \mathbb Z$, in other words, a two-dimensional 0-1 block (light = 1, no light = 0). What you need to do is to make a night reconnaissance, examine the configuration of the lights already existing in your world and find a finite configuration of lights that does not occur anywhere. This block is going to be your marker. When you apply this very configuration to mark the landing place, and inform the pilot about the pattern, he will then be able to determine the landing spot. Let me mention, that in order to be able to ``apply the marker'', some kind of specification property of the admissible 0-1 patterns is assumed. When choosing the marker you must also make sure that by placing it you will not create a ``false marker'' (indicating a wrong place) consisting of some lights of your marker combined with some of the formerly existing lights. I will explain what sort of difficulties occur when the $\mathbb Z\times \mathbb Z$ lattice is replaced by an abstract countable amenable group and how they are overcome.