empty set is never fat, which implies that such dimension must exist (Proposition \[good-dimension-pro\]).

The last step is to deduce from the links of the non-fat $(j-1)$-faces that there exist many expanding $(i+1)$-faces. We already know, by the existence of many fat $j$-faces in these links, that a large part of $W$ is seen in them (Proposition \[container-bound-pro\]). It is left to show that many of the $(i+1)$-faces in these links contain at least one $i$-face which is not in $W$ (Proposition \[full-faces-upper-bound-pro\]).

For that we define [*full faces*]{} for any dimension of the complex, where a $k$-face is full if it contains only fat $(k-1)$-faces. By this definition it follows that a full $k$-face is seen in the link of any $(k-2)$-face as an edge between two fat vertices. So when considering the link of a non-fat $(k-2)$-face, which by definition contains only a few fat vertices, by the skeleton expansion it contains only a few full $k$-faces. Thus, the number of the full $(j+1)$-faces in the links of the non-fat $(j-1)$-faces is very small. Then again, by the skeleton expansion and the maximality of $j$ we can deduce that for any $k>j+1$ most of the full $k$-faces contain 