Although Theorem \[th:general\] gives infinitely many linear converse bounds on $R^*$, the region of the memory-rate pair $(M,R^*)$ characterized by Theorem \[th:general\] has a simple shape with finite corner points. Specifically, by applying the arguments used in the proof of Theorem \[th:gconst\], one can show that the exact bounded region given by Theorem \[th:general\] is bounded by the lower convex envelop of points $\{(\frac{N-\ell+1}{s}, \frac{s-1}{2}+\frac{\ell(\ell-1)}{2s})\ |\ s\in\{1,...,J\}, \ell\in\{1,...,s\} \}\cup \{({0,J})\}$, where $J=\min\{N,K\}$.

For the case of large $N$, we can exactly characterize the values of $R^*$ and $R^*_{\textup{ave}}$ for $K\leq 5$. We formally state this result in the following theorem:

\[th:exact5\] For a caching system with $K$ users, a database of $N$ files, and a local cache size of $M$ files at each user, we have $$R^*=R^*_{\textup{ave}}=R_{\textup{u}}(N,K,r)$$ for large $N$ (i.e., $N\rightarrow+\infty$) when $K\leq 5$, where $R_\textup{u}(N,K,r)$ is defined in Definition \[def\].[^13]

As discussed in [@maddah-ali13], the special case of large $N$ is important to handle asynchronous demands. More specifically, [@maddah-ali13] showed that asynchronous demands can be handled by splitting each file into many subfiles, and delivering concurrent subfile requests using the optimum caching schemes. In this case, we essentially need to solve the caching problem when the number of 