Let $A_1,\dotsc,A_n$ be events in a probability space, and let $N = \sum_{i=1}^n \mathbf{1}_{A_i}$ be the random number of events that occur. For a fixed value $k \in \{1,\dotsc,n\}$, what can be said about the quantity $$ p_k = \mathbb{P}[ N \ge k ] $$ if all that is known is the number $n$ and upper bounds on the single probabilities $\mathbb{P}[A_i]$ and pairwise probabilities $\mathbb{P}[A_i \cap A_j]$?

A simple approach is Chebyshev's inequality, since the first two moments of $N$ can be expressed in terms of these probabilities. Focusing for simplicity on the symmetric case ($\mathbb{P}[A_i]$ is the same for all $i$, $\mathbb{P}[A_i \cap A_j]$ is the same for all $i\ne j$), a simple calculation gives
$$ \mathbb{E}[N] = n \mathbb{P}[A_1] $$
$$ \mathbb{E}[N^2] = n \mathbb{P}[A_1] + (n^2 - n) \mathbb{P}[A_1 \cap A_2]. $$
However, we then have $\mathrm{Var}[N] = \mathbb{E}[N^2] - \mathbb{E}[N]^2$, so it appears that getting an upper bound on $p_k$ via this approach requires an upper bound on $\mathbb{E}[N^2]$ and *both upper and lower bounds on $\mathbb{E}[N]$*, the latter being used to upper bound the variance (the trivial upper bound of $\mathbb{E}[N^2]$ does not suffices for my purposes).

What if we only have have access to upper bounds? In particular, can we obtain a good upper bound on $p_k$ if we only know that $\mathbb{P}[A_i] \le \alpha$ and $\mathbb{P}[A_i \cap A_j] \le \alpha^2$ for some $\alpha > 0$?

In the case that $k=1$, $p_k$ is simply the probability of a union of events, so we may use tools such as the union bound, the inclusion-exclusion principle, and de Caen's bound (http://dl.acm.org/citation.cfm?id=253949) to get useful upper and lower bounds. It is unclear whether these have analogs for higher values of $k$.