超級混合圖是一個 H = (X,C,D) 的表示法，其中X是代表點集合，而C和D是X的部分子集合，稱為邊。一個嚴格k種顏色可著色法指的是由X的點集對應到{1,2,…,k}的一種關係，其中C代表每一個C邊至少有兩個點同色，而D代表每一個D邊至少有兩個點不同色。C和D都有可能是空集合。假如超過(少於)k並沒有可著色的方法數，則k稱為最大著色數(最小著色數)。而H的每個邊都恰好有r個點則稱為r均勻超級混合圖。
對於r均勻C超級混合圖，如果限定了最大著色數大於等於k的話，則將會改變最大著色數的邊數。如果要找出滿足此條件的最大著色數的最大的邊數，我們主要區分成三種不同的情形來討論，分別是r比k大、r比k小和r = k。

A mixed hypergraph is a triple H = (X, C,D), where X is the vertex set, and each of C,D is a list of subsets of X. A strict k-coloring is a onto mapping from X to {1,2, . . . , k} such that each C ∈ C contains two vertices have a common value and each D ∈ D has two vertices have distinct values. Each of C,D may be empty. The maximum(minimum)
number of colors over all strict k-colorings is called the upper(lower) chromatic number of H and is denoted by χ^¯(H)(χ(H)). If a hypergraph H has no multiple edges and all its
edges are of size r, then H is called an r-uniform hypergraph. We want to find the maximum number of edges for r-uniform C-hypergraph of order n with the condition χ^¯(H) ≥ k, where k is fixed. We will solve this problem according to three different cases, r < k, r = k and r > k.

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