If \(\mathcal{T}\) is a collection of trees on \(\omega^\omega\), then we define the tree ideal \(t_0\) as a collection of these \(X\subset \omega^\omega\) such that each \(T\in\mathcal{T}\) has a subtree \(S\in\mathcal{T}\) which shares no branches with \(X\). We will be interested in the cofinalities of tree ideals. Building on the work of Brendle, Khomskii, and Wohofsky, we will analyse the condition called "Incompatibility Shrinking Property", which implies that \(cof(t_0)>2^\omega\). We will investigate under which assumptions this property is satisfied for two types of trees. These types are Laver and Miller trees which split positively according to some fixed ideal on \(\omega\). (Joint work with Arturo Martinez Celis.)
