Weakened Ramsey numbers are a natural generalization of the definition of a tcolored Ramsey number Rt (G), defined to be the least natural number p such that every t-coloring of the edges in the complete graph Kp has a monochromatic subgraph isomorphic to G. For s < t, one can define Rt s(G) to be the least natural number p such that every t-coloring of the edges in Kp contains a subgraph isomorphic to G that is spanned by edges in at most s colors. It follows that Rt 1(G) = Rt (G), but few explicit values are known for values of s > 1. The goals of this article are two-fold. First, we show how the work of Su, Li, Luo, and Li done in 2002 can be used to derive new lower bounds for fifteen weakened Ramsey numbers of the form R3 2(Kn). Then we turn our attention to the analogue of weakened Ramsey numbers in the setting of r-uniform hypergraphs, proving some explicit and general bounds for such numbers.
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