Linear upper bounds for local Ramsey numbers
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A coloring of the edges of a graph is called alocal k-coloring if every vertex is incident to edges of at mostk distinct colors. For a given graphG, thelocal Ramsey number, r loc k (G), is the smallest integern such that any localk-coloring ofK n , (the complete graph onn vertices), contains a monochromatic copy ofG. The following conjecture of Gyárfás et al. is proved here: for each positive integerk there exists a constantc = c(k) such thatr loc k (G) ≤ cr k (G), for every connected grraphG (wherer k (G) is theusual Ramsey number fork colors). Possible generalizations for hypergraphs are considered.
KeywordsComplete Graph Distinct Color Ramsey Number Monochromatic Copy Number Fork
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