We will show that the Nash-Williams and Galvin’s theorems are in fact generalizations of the infinite Ramsey theorem. In other words, it is possible to prove the infinite Ramsey theorem by assuming that the Nash-Williams or the Galvin’s theorem is true. The Nash-Williams and Galvin’s theorems are results about the monochromatic sets of a given family F of finite subsets of N. For the Galvin’s theorem the definition of monochromatic will change abruptly since it will no longer depend on a coloring of F , but only on F . Thus, in this case the term homogenous makes more sense.

Camilo Nuñez - [email protected]