## Thesis etd-09022019-214057 |

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Thesis type

Tesi di laurea magistrale

Author

BERTOLUCCI, GIACOMO

URN

etd-09022019-214057

Thesis title

Finite colorings, partition regularity and families of large sets

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Prof. Di Nasso, Mauro

**controrelatore**Mamino, Marcello

Keywords

- colorings
- combinatorial number theory
- Furstenberg families
- large sets
- Moreira's theorem
- partition regular
- piecewyse syndetic
- ramsey families
- Ramsey theory
- syndetic
- thick

Graduation session start date

20/09/2019

Availability

Full

Summary

We will start by introducing some variants of the concept of partition regularity and by defining the families of syndetic, thick and piecewise syndetic sets, of which we will give some basic properties. Then we will display a theorem by J. Moreira concerning the existence of monochromatic configurations of a special type, based on a result by V. Bergelson and N. Hindman; we will give a combinatorial proof obtained from the original dynamical proof using a sketch by Moreira himself.

We will proceed introducing PS-chains and giving some of their properties. We will use them in order to state and prove a first generalization of Moreira's theorem, which adds some information about the largeness of the set of monochromatic configurations in terms of piecewise syndetic sets. We will give a second generalization of the theorem, replacing the family of piecewise syndetic sets with a more general class of "families of large sets". Then we will exhibit some examples which show that the family of syndetic sets does not have some of the properties which the family of piecewise syndetic sets instead has.

Next we will show a possible application of our generalizations of Moreira's theorem, proving the existence of a large set of monochromatic solutions for two particular classes of Diophantine equations, one of which has already been identified by Moreira.

We will generalize the concepts of syndeticity, thickness and piecewise syndeticity to subsets of a generic semigroup, and we will show the (lack of) relations between these families and the corresponding families of chains. Then we will introduce Furstenberg families and the main operations on them, giving some basic properties. Finally, we will introduce the tensor product of families, an operation which is strictly linked to chains, displaying some of its properties and showing its behavior with respect to other operations, proving that, under certain hypotheses, it preserves filters, ultrafilters and partition (quasi-)regularity.

In the Appendices, we will display some sketches about a possible combinatorial proof for the result by Bergelson and Hindman.

We will analyze the properties of piecewise syndetic sets in more depth, displaying some results which link the (piecewise) syndeticity of the sum of two sets with their densities and exposing two characterizations of piecewise syndetic sets, one in terms of finite embeddability of a syndetic set, the other concerning a particular class of ultrafilters.

We will show the equivalence between being a partition (quasi-)regular family, having a filter as dual family and (for Furstenberg families) being a union of ultrafilters.

We will proceed introducing PS-chains and giving some of their properties. We will use them in order to state and prove a first generalization of Moreira's theorem, which adds some information about the largeness of the set of monochromatic configurations in terms of piecewise syndetic sets. We will give a second generalization of the theorem, replacing the family of piecewise syndetic sets with a more general class of "families of large sets". Then we will exhibit some examples which show that the family of syndetic sets does not have some of the properties which the family of piecewise syndetic sets instead has.

Next we will show a possible application of our generalizations of Moreira's theorem, proving the existence of a large set of monochromatic solutions for two particular classes of Diophantine equations, one of which has already been identified by Moreira.

We will generalize the concepts of syndeticity, thickness and piecewise syndeticity to subsets of a generic semigroup, and we will show the (lack of) relations between these families and the corresponding families of chains. Then we will introduce Furstenberg families and the main operations on them, giving some basic properties. Finally, we will introduce the tensor product of families, an operation which is strictly linked to chains, displaying some of its properties and showing its behavior with respect to other operations, proving that, under certain hypotheses, it preserves filters, ultrafilters and partition (quasi-)regularity.

In the Appendices, we will display some sketches about a possible combinatorial proof for the result by Bergelson and Hindman.

We will analyze the properties of piecewise syndetic sets in more depth, displaying some results which link the (piecewise) syndeticity of the sum of two sets with their densities and exposing two characterizations of piecewise syndetic sets, one in terms of finite embeddability of a syndetic set, the other concerning a particular class of ultrafilters.

We will show the equivalence between being a partition (quasi-)regular family, having a filter as dual family and (for Furstenberg families) being a union of ultrafilters.

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