• central sets
• combinatorics
• exponential configurations
• Ramsey theory
• ultrafilters

This thesis is about central sets, special subsets of the natural numbers with a rich combinatorial structure, described by the so-called Central Sets Theorem. As a consequence, central sets contain arbitrarily long arithmetic progressions and are additively big, namely there exists an infinite sequence whose finite sums are included in the set; what is more, it generalizes important results of Ramsey Theory.
Central sets were introduced by H. Furstenberg in 1981 in the context of topological dynamics, but later V. Bergelson observed that they can be defined equivalently as members of specific ultrafilters.
In the final part we use them to prove a new result in Ramsey Theory. In 2018 J. Sahasrabudhe proved that, given a finite coloring of the natural numbers, there exist arbitrarily long sequences such that the finite products and certain finite exponentiations of members from them are monochromatic.
Our new result states that, given a finite coloring of the natural numbers, there exists an infinite sequence such that certain finite exponentiations of elements from the sequence are monochromatic. So we find an infinite sequence, no more arbitrarily long but finite ones and, on the other hand, we do not include finite products, like Sahasrabudhe did instead.