Thick, syndetic, and piecewise syndetic subsets of Fräıssé structures
نویسنده
چکیده
We define and undertake a systematic study of thick, syndetic, and piecewise syndetic subsets of a Fräıssé structure. Each of these collections forms a family in the sense of Akin and Glasner [AG], and we define and study ultrafilters on each of these families, paying special attention to ultrafilters on the thick sets. In the process, we generalize many results of Bergelson, Hindman, and McCutcheon [BHM]. We also discuss some abstract questions about families implicit in the work of Akin and Glasner.
منابع مشابه
Density near zero
Let $S$ be a dense subsemigroup of $(0,+infty)$. In this paper, we state definition of thick near zero, and also we will introduce a definition that is equivalent to the definition of piecewise syndetic near zero which presented by Hindman and Leader in [6]. We define density near zero for subsets of $S$ by a collection of nonempty finite subsets of $S$ and we investigate the conditions un...
متن کاملAlmost Disjoint Large Subsets of Semigroups
There are several notions of largeness in a semigroup S that originated in topological dynamics. Among these are thick , central , syndetic and piecewise syndetic. Of these, central sets are especially interesting because they are partition regular and are guaranteed to contain substantial combinatorial structure. It is known that in (N,+) any central set may be partitioned into infinitely many...
متن کاملMonochromatic Forests of Finite Subsets of N
It is known that if N is finitely colored, then some color class is piecewise syndetic. (See Definition 1.1 below for a definition of piecewise syndetic.) We generalize this result by considering finite colorings of the set of all finite subsets of N . The monochromatic objects obtained are “d-copies” of arbitrary finite forests and arbitrary infinite forests of finite height. Van der Waerden’s...
متن کاملUsing Βs in Combinatorics and Dynamical Systems (survey)
1. Basic properties of ultrafilters 2. Applications of ultrafilters. The space βS 3. Semigroups: basic facts and examples 4. βS as a semigroup 5. Hindman’s finite sums theorem 6. Semigroups: ideals and minimal idempotents 7. The Hales-Jewett theorem 8. Large sets: thick, central, IP-, and syndetic sets 9. Large sets: piecewise syndetic sets 10. Dynamical systems: basic notions 11. βS as a dynam...
متن کاملHigh density piecewise Syndeticity of Product Sets in Amenable Groups
M. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = Z, this was first proven by R. Jin. We prove a quantitative ver...
متن کامل