Let $N_1$, $N_2$, and $k$ be positive integers. Let $V_1$ and $V_2$ be finite sets with $|V_i| = N_i \ge 1$. Consider a bipartite graph $G=(V_1,V_2,E)$ constructed as follows. For every $x \in V_1$, sample a $k$-element subset $S_x$ of $V_2$ with replacement, and connect $x$ to every $y \in S_x$. Thus, the degree of every $x$ is exactly $k$.
Now, assume $N_2 \ll N_1 \to \infty$, and $k$ is fixed. For $\theta,\alpha,\gamma \in (0,1)$, define $p=p(k,\theta,\alpha,\gamma) \in [0,1]$ by
$$ p := \lim\mathbb P\Big(\forall X \subseteq V_1\text{ with }|X| = \theta N_1,\,\exists Y \subseteq V_2\text{ with }|Y| \ge (1-\alpha) N_2\text{ s.t }\max\mathrm{deg}_X(Y) \le \gamma k \theta N_1\Big), $$ where $\max\mathrm{deg}_X(Y) := \sup_{y \in Y}\mathrm{deg}_X(y)$ and $\mathrm{deg}_X(y)$ is the number of edges between the vertex $y$ and the set $X$.
I'm interested in phase transitions in $p$ as a function of $k$, $\theta$, $\alpha$, and $\gamma$.
Question 1. Is there an open region $R$ in $(k,\theta,\alpha,\gamma)$-space such that such that $p=1$ on $R$, and $p=0$ on the interior of the complement of $R$ ?
Question 2. What kinds of tools can be used to attack such problems ? Are results like Gale-Ryser or Erdos-Gollai applicable here ?
Partial solution using the "probabilistic method"
Let $N \ge 0$ be the number of subsets $Z$ of $V_2$ of size $\alpha N_2$ such that $\max\mathrm{deg}_X(Z) \ge \gamma k \theta N_1$. We will carve-out an open region of space on which $N = o(1)$; this implies $p=1$ on this region.
To this end, observe that the expected value of $N$ can be upper-bounded like so $$ \overline N \le {N_1\choose\theta N_1} \cdot {N_2\choose \alpha N_2}\cdot f(k\theta N_1,\gamma,\alpha), $$ where $f(n,\gamma,\alpha) := \mathbb P(Bin(n,\alpha) \ge \gamma n) = \sum_{i=\lfloor \gamma n\rfloor}^n {n\choose i}\alpha^i(1-\alpha)^{n-i}$.
Let $H(\theta)$ be the binary entropy of the Bernoulli distribution $B_\theta$ with parameter $p$ and let $D(\gamma \mid \alpha)$ be the binary KL divergence between $B_\gamma$ and $B_\alpha$. Using the the well-known bounds
- ${n\choose pn} \asymp 2^{H(p)n}$ for large $n$,
- $f(n,\gamma,\alpha) \le 2^{-D(\gamma ||\alpha)n}$ (for example, see https://mathoverflow.net/a/404642/78539), we have $$ \begin{split} \frac{\log \overline N}{N_1} &\le H(\theta) + H(1/2) \frac{N_2}{N_1} - D(\gamma || \alpha)k\theta + o(1)\\ &= H(\theta) - D(\gamma || \alpha)k\theta + o(1) \end{split} $$ where we've used the fact that ${N_2\choose \alpha N_2} \le {N_2\choose N_2/2} \le 2^{H(1/2)N_2}$ and also $N_2/N_1 = o(1)$ since $N_1 \gg N_2$. Thus, if the RHS is strictly negative, that is if we're in the open region $R$ given by the inequality $$ \frac{H(\theta)}{\theta} - kD(\gamma ||\alpha) \lt 0. $$
In this region, we must have $\overline N \le 2^{-\Omega(1) N_1}$ in the limit $N_1 \to \infty$. We deduce that $p \ge \mathbb P(N = 0) \to 1$, i.e $p=1$ on the region $R$.
