We extend results of Gyárfás and Füredi on the largest monochromatic component in r-colored complete k-uniform hypergraphs to the setting of random hypergraphs. We also study long monochromatic loose cycles in r-colored random hypergraphs. In particular, we obtain a random analog of a result of Gyárfás, Sárközy, and Szemerédi on the longest monochromatic loose cycle in 2-colored complete k-uniform hypergraphs.

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Given a graph G, we say a k-uniform hypergraph H on the same vertex set contains a Berge-G if there exists an injection φ : E(G) → E(H) such that e ⊆ φ(e) for each edge e ∈ E(G). A hypergraph H is Berge-G-saturated if H does not contain a Berge-G, but adding any edge to H creates a Berge-G. The saturation number for Berge-G, denoted sat k (n, Berge-G) is the least number of edges in a k-uniform hypergraph that is Berge-G-saturated. We determine exactly the value of the saturation numbers for Berge stars. As a tool for our main result, we also prove the existence of nearly-regular k-uniform hypergraphs, or k-uniform hypergraphs in which every vertex has degree r or r − 1 for some r ∈ Z, and less than k vertices have degree r − 1.

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