In the paper, \(k\)-local and \(k\)-mean colorings of graphs and hypergraphs are studied. Given an edge-coloring \(f\) and a vertex \(v\), \(\alpha_ f(v)\) denotes the number of distinct colors that appear on the edges incident with \(v\). A coloring \(f\) is \(k\)-local if for every vertex \(v\), \(\alpha_ f(v) \leq k\). A coloring \(f\) is \(k\)-mean if \((1/n) \sum_ v \alpha_ f(v) \leq k\), where \(n\) is the number of vertices in a graph (hypergraph). The paper contains several Ramsey-type results on colorings of both types. In particular, the density theorem is proved for \(k\)-mean colorings. It says that in each \(k\)-mean coloring of a graph with average degree at least \(d\), there is a monochromatic subgraph with the average degree at least \(d/k\). Ramsey numbers of complete graphs for 2-local and 2-mean colorings are studied. In some cases, exact results are obtained. Relations between \(k\)-mean Ramsey numbers and classical Ramsey numbers are also studied.