of Theorem \[main-thm-1-informal\].

We need to show that for any dimension $i$ and any non-empty $i$-cochain with norm up to $1/2$ there are many expanding faces.

Consider an arbitrary $i$-cochain $W$. From a very high level point of view, the proof can be divided into three steps:

1.  Mark “selected” faces in dimensions $i,i-1,\dotsc,-1$.

2.  Find a “good” dimension $j \le i$.

3.  “Climb up” from dimension $j$ to deduce many expanding faces in dimension $i+1$.

In more details, we start by marking all of the $i$-faces which are in $W$. Then we mark $(i-1)$-faces, where the rule for marking a face is if many (above some threshold) of the $i$-faces which contain it are marked. We continue this way down to the lowest dimension of the complex, where each dimension is marked with regard to one dimension above. We term these marked faces as [*fat faces*]{} since each marked face contains in its link many $i$-faces which are in $W$.

The next step is to look for the highest dimension $j \le i$ with the property that many fat $j$-faces contain many non-fat $(j-1)$-faces. By setting a large enough threshold for marking faces we can make sure that the 