RELAZIONI DI PARTIZIONE PER ORDINALI NUMERABILI

In 1930, Ramsey published his famous theorem and this result generated Ramsey theory and the study of partition relations. A partition ordinal is an ordinal \alpha such that for any coloring on the edges of the complete graph with order type \alpha with 2 colors, there exists a clique with order type \alpha colored with color 0, or there is a triangle in color 1. In this thesis we study for which countable ordinals this relation holds. In chapter 1 we deal with Ramsey's theorems: infinite Ramsey's theorem and finite Ramsey's theorem. We see some important partition relations for cardinals greater than \omega and next we give a brief presentation of finite Ramsey theory. In chapter 2 we study classical results on partition relations for countable ordinals, that were found between 1957 and 1975. Thanks to Specker, Galvin and Larson we know that \omega^\alpha is a partition ordinal if \alpha \in {0,2,\omega}, and that \omega^n is not a partition ordinal if n in a natural number greater than 2. In chapter 3 we present Schipperus' results. He proved that \omega^{\omega^\beta} is a partition ordinal if \beta is indecomposable or if it is sum of two indecomposable ordinals. In chapter 4 we present the negative partition relations founded by Schipperus (\omega^{\omega^\beta} is not a partition ordinal if \beta is sum of four or more indecomposable ordinals) and a summary of main theorems and open problems.