Sealing Kurepa Trees in ZFC

A class of trees may be defined using shared properties: height, width, cardinality of  the branch set, etc. For a tree T within a model M we ask if we may add a branch (or branches) to the model using some forcing notion without destroying said properties. If we cannot, we say the tree is sealed. A natural followup question is, under what conditions can we build a forcing notion which does not harm the properties of the tree, and seals it? In this lecture I show that given a Kurepa tree in M without any other assumptions on the model (except that it is a model of ZFC), we can construct a proper forcing-notion which forces a tree to be sealed for every forcing-notion from the ground model, without collapsing \aleph_2 and \aleph_1 .