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The centralized coded caching scheme is a technique proposed by Maddah-Aliand Niesen as a solution to reduce the network burden in peak times in awireless system. Later Yan et al. reformulated the problem as designing acorresponding placement delivery array, and proposed two new schemes from thisperspective. These schemes above significantly reduce the transmission rate$R$, compared with the uncoded caching scheme. However, to implement the newschemes, each file should be cut into $F$ pieces, where $F$ increasesexponentially with the number of users $K$. Such constraint is obviouslyinfeasible in the practical setting, especially when $K$ is large. Thus it isdesirable to design caching schemes with constant rate $R$ (independent of $K$)as well as small $F$.  In this paper we view the centralized coded caching problem in a hypergraphperspective and show that designing a feasible placement delivery array isequivalent to constructing a linear and (6, 3)-free 3-uniform 3-partitehypergraph. Several new results and constructions arise from our novel point ofview. First, by using the famous (6, 3)-theorem in extremal combinatorics, weshow that constant rate caching schemes with $F$ growing linearly with $K$ donot exist. Second, we present two infinite classes of centralized coded cachingschemes, which include the schemes of Ali-Niesen and Yan et al. as specialcases, respectively. Moreover, our constructions show that constant ratecaching schemes with $F$ growing sub-exponentially with $K$ do exist.

Centralized coded caching schemes: A hypergraph theoretical approach