• combinatorics
• dynamical systems
• intersective sets
• mathematical logic
• non-standard analysis
• non-standard methods
• Ramsey theory
• recurrence
• topology
• ultrafilters

During the last 50 years, the connection between problems in Ramsey Theory and dynamical systems was deeply studied. In fact, many problems about partition regularity (like Van der Waerden's theorem) found a correspondence in the setting of topological dynamical systems. On the other hand, density problems (like Szemerédi's theorem) fit more in the setting of measure preserving systems. These two settings are philosophically similar, though not identical. Specifically, in both contexts we can define a concept of recurrent sets of natural numbers: it is possible to prove that recurrent sets are not the same in the two contexts. The objective of the thesis is study the difference of the two types of recurrent sets: we are interested in the study of intersective sets, a notion which allows us to transform recurrence in a purely combinatorial property. In particular, we will show some interesting combinatorial properties of some intersective sets, using non-standard methods.