Abstract:

The thesis consists of four chapters and one appendix. The relation between trees and ordering types, especially the relation between treesubtree and the typesubtype are considered in Chapter 1. By using Jensens’s principle, Aronszajn’s tree which does not contain any Aronszajn’s subtree and Cantor’s subtree are constructed. Moreover, it is shown that in the model ZFC+GCH each ω_2 Aronszajn’s tree contains Aronszajn’s and Cantor’s subtree. In the first part of Chapter 2 the problem of the existence of Boolean algebras which have nontrivial automorphisms and endomorptisms are studied. It is shown that for each cardinal k, k>ω, there are exactly 2^k types of isomorphic Boolean algebras without nontrivial automorphisms. In the second part of that chapter the problem of isomorphism and automorhism of ω_1trees is studied. It is shown that there are 2^ω1 types of isomorphic total rigid Aronszajn’s trees, so one Aronszajn’s tree does not have any nontrivial automorphism. Several problems of the partition relations of cardinal numbers are solved in Chapter 3. The appendix contains the proof of the property that in ZFC the σdense partial ordered set of power ω_1 does not exist. It is shown that in ZFC there is not any linearly ordered topological space with weight less or equal ω_1 which satisfies Kurepa’s generalization of the notion of separable topological space. It is also shown that if ¬ω Kurepa’s hypothesis + Martin’s axiom + ¬Continuum hypothesis is assumed, then each perfect normal non  Arhimedian space whose weight is ω1 is measurable. 