Abstract
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 k loc (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 k loc (G) ≤ cr k (G), for every connected grraphG (wherer k (G) is theusual Ramsey number fork colors). Possible generalizations for hypergraphs are considered.
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Gyárfás, A., Lehel, J., Schelp, R.H., Tuza, Zs.: Ramsey numbers for local colorings. Graphs and Combinatorics (to appear)
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On leave from the Institute of Mathematics, Technical University of Warsaw, Poland.
While on leave at University of Louisville, Fall 1985.
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Truszczynski, M., Tuza, Z. Linear upper bounds for local Ramsey numbers. Graphs and Combinatorics 3, 67–73 (1987). https://doi.org/10.1007/BF01788530
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DOI: https://doi.org/10.1007/BF01788530