Linear upper bounds for local Ramsey numbers

A coloring of the edges of a graph is called a local k-coloring if every vertex is incident to edges of at most k distinct colors. For a given graph G, the local Ramsey number, r_loc^k(G), is the smallest integer n such that any local k-coloring of K_n, (the complete graph on n vertices), contains a monochromatic copy of G. The following conjecture of Gyárfás et al. is proved here: for each positive integer k there exists a constant c = c(k) such that r_loc^k(G) ≤ cr^k(G), for every connected grraph G (where r^k(G) is the usual Ramsey number for k colors). Possible generalizations for hypergraphs are considered.