I apologize for the fairly unspecific title. I am working through a paper with a technical lemma that I do not understand.
Let $H$ be a $k$-uniform hypergraph.
Definition. $s(H)$ is the smallest subset of hyperedges in $H$ that does not have a set of distinct representatives (a set of vertices with a one-to-one correspondence to edges).
Hall's Theorem tells us that this an edge-set has this property if and only if
Definition. The boundary of $H$, denoted $\delta H$, is the set degree 1 vertices in $H$.
Definition. The sub-critical expansion of $H$, denoted $e(H)$, be $$ e(H) = \max_{s \leq s(H)} \min\{|\delta G| : G \subseteq H, s / 2 \leq |G| < s\}. $$
I imagine many of these definitions are standard in the study of expander families, but I'm not too familiar with this area.
The simple step that I do not understand is the following upper bound. For a random $k$-uniform hypergraph $H$ ($m$ $k$-edges choses uniformly with replacement),
$$ \Pr[e(H) \leq s] \leq \sum_{r = s / 2}^s \binom{n}{r} \Pr[B(m, (r / n)^k) \geq 2r / (k + \epsilon)] $$
Where $B(m, p)$ is given by the binomial distribution from $m$ trials and probability $p$ of success. The author explains that if the density of a $k$-uniform hypergraph bounded by $2 / k - \epsilon$, then the average degree is bounded above by 2 and the number of elements in its boundary must be a constant fraction of the total number of variables. I believe this fact is straightforward to prove.
So far I can see that he is saying that the probability that $e(H) \leq s$ is bounded above by the probability that there is some set vertices of size between $s / 2$ and $s$ with an edge density greater than $2 / (k + \epsilon)$, so the boundary might be less than a constant fraction of the vertices. But I have a hard time interpreting this is a more useful way. In particular, I am confused about the fact that the definition of expansion is given in terms of sizes of edge-sets, but the union bound in this expression is taken over vertices. I imagine there is something quite simple that I am missing.
For those interested, this bound is from a lemma in the book Computational Complexity Theory by Steven Rudich and Avi Wigderson, in the section on Proof complexity. It is a standard lemma for proving resolution length lower bounds for random CNF formulas by reducing to width lower bounds. I have used mostly the same variable names as given in their proof.