files (i.e., the subfiles) is large, but the fraction of files that can be stored at each user is fixed. In this paper, we completely characterize this tradeoff for systems with up to $5$ users, for both peak rate and average rate, while in prior works, this tradeoff has only been exactly characterized in two instances: the single-user case [@maddah-ali12a] and, more recently, the two-user case [@tian2016symmetry].

Although Theorem \[th:exact5\] only consider systems with up to $5$ users, the converse bounds used in its proof also tightly characterize the minimum communication rate in many cases even for systems with more than $5$ users. For both peak rate and average rate, we can show that more than half of the convex envelope achieved by [@yu2016exact] are optimal for large $N$ (e.g., see Lemma \[lemma:2\] for peak rate).

To prove Theorem \[th:exact5\], we state the following Theorem, which provides tighter converse bounds on $R^*$ for certain values of $N$, $K$, and $M$.

\[th:2\] For a caching system with $K$ users, a database of $N$ files, and a local cache size of $M$ files at each user, $R^*$ is lower bounded by $$\begin{aligned}
\label{eq:thm4}
  R^*\geq \begin{cases}
               \frac{2K-n+1}{n+1}-\frac{K(K+1)}{n(n+1)}\cdot \frac{M}{N}\ \ \ \ \ \  