Weakly reflecting graph properties

L. Soukup formulated an abstract framework in his introductory paper for proving theorems about uncountable graphs by subdividing them by an increasing continuous chain of elementary submodels. The applicability of this method relies on the preservation of a certain property (that varies from problem to problem) by the subgraphs obtained by subdividing the graph by an elementary submodel. He calls the properties that are preserved ``well-reflecting''. The aim of this paper is to investigate the possibility of weakening of the assumption ``well-reflecting'' in L. Soukup's framework. Our motivation is to gain better understanding about a class of problems in infinite graph theory where a weaker form of well-reflection naturally occurs

Sealed Kurepa Trees

In this paper we investigate the problem of the distributivity of Kurepa trees. We show that it is consistent that there are Kurepa trees and for every Kurepa tree there is a small forcing notion which adds a branch to it without collapsing cardinals. On the other hand, we derive a proper forcing notion for making an arbitrary Kurepa tree into a non-distributive tree without collapsing $\aleph_1$ and $\aleph_2$