The number of Seymour vertices in random tournaments and digraphs Maybe this is useful?

Also, for your inequality in Lemma 2, maybe put a comma after $p\geq 1$, because I first read it as $p=0$ or ($p\geq 1$ and $p_1,…,p_r\in [0,p]$), not ($p=0$ or $p\geq 1$) and $p_1,…,p_r\in [0,p]$ as I later understood it. I was wondering if this inequality follows from an existing well-known inequality, or maybe just what your intuition was for why this should be true.

I started to come to that realization after I asked the question. That is, of course we can count the edges inside the set $U$ and the edges between $U$ and $W$, but we might as well just count the edges inside $W$ since that is where the problem will lie.

Thanks. This is exactly the type of thing I was looking for. I'm currently trying to figure out if we can just bound the number of edges in each $G_i$ using Berge/Tutte as you did, and then add them all up and show that it is less than $\binom{n}{2}$. Or is it vital to the proof that we must first define the set $W$ and upper bound the number of edges inside $W$ while lower bounding the size of $W$?