Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory (Paperback, 2019)

Preface.- Notation and Conventions.- Part I Preliminaries.- 1. Ultrafilters.-2. Nonstandard analysis.- 3. Hyperfinite generators of ultrafilters.- 4. Many stars: iterated nonstandard extensions.- 5. Loeb measure.- Part II Ramsey Theory.- 6. Ramsey’s Theorem.- 7. The theorems of van der Waerden and Hales-Jewett.- 8. From Hindman to Gowers.- 9. Partition regularity of equations.- Part III Combinatorial Number Theory.-10 Densities and structural properties.- 11. Working in the remote realm.- 12. Jin’s Sumset Theorem.- 13. Sumset configurations in sets of positive density.- 14. Near arithmetic progressions in sparse sets.- 15. The interval measure property.- Part IV Other topics.-16. Triangle removal and Szemeredi regularity.- 17. Approximate groups.- Part V Appendix.-A. Foundations of nonstandard analysis.-References.-Index.