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A Gallai-coloring (G-coloring) is a generalization of 2-colorings of edges of complete graphs: a G-coloring of a complete graph is an edge coloring such that no triangle is colored with three distinct colors. Here we extend some results known earlier for 2-colorings to G-colorings. We prove that in every G-coloring of Kn there exists each of the following: 1. a monochromatic double star with at least 3n+1 4 vertices; 2. a monochromatic subgraph H such that all pairs of X ½ V (Kn) are at distance at most two in H where jXj ¸ d3n 4 e; 3. a monochromatic diameter two subgraph with at least d3n 4 e vertices. We also investigate Ramsey numbers of graphs in G-colorings with a given number of colors. For any graph H let RG(r;H) be the minimum m such that in every G-coloring of Km with r colors, there is a monochromatic copy of H. We show that for fixed H, RG(r;H) is exponential in r if H is not bipartite; linear in r if H is bipartite but not a star; constant (does not depend on r) if H is a star (and we determine its value). Somewhat surprisingly, RG(r;K3) can be determined exactly.