Abstract
The main result provides a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for natural numbers due to Milliken–Tylor, Deuber–Hindman, Bergelson–Hindman, for combinatorial covering properties due to Scheepers and Tsaban, and local properties in function spaces due to Scheepers. To this end, we use idempotent ultrafilters in the Čech–Stone compactifications of discrete infinite semigroups and topological games. The research is motivated by the recent breakthrough work of Tsaban about colorings and the Menger covering property.
| Original language | English |
|---|---|
| Article number | 108595 |
| Journal | Topology and its Applications |
| Volume | 335 |
| DOIs | |
| State | Published - 1 Aug 2023 |
| Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2023
Funding
I would like to thank Boaz Tsaban, who introduced me to this topic, for encouragement to continue his work and his great impact to my research. I would like to thank Marion Scheepers, who draw my attention that for a space, if A∪B∈Ω, then one of the families A or B is in Ω. I am grateful to the anonymous referee for careful reading of the manuscript and all corrections.
| Funders | Funder number |
|---|---|
| Boaz Tsaban | |
| Marion Scheepers |
Keywords
- Combinatorial covering properties
- Finite colorings
- Infinite topological games
- Local properties in function spaces
- Menger's property
- Rothberger's property
- Selection principles
- Semigroups
- Čech–Stone compactification