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We prove that for $n$ large and every $k$ with $k\\le 2^{n/4300}$, the number of Gallai colorings of $K_n$ that use at most $k$ given colors is $(\\binom{k}{2}+o_n(1))\\,2^{\\binom{n}{2}}$. Our result is asymptotically best possible and implies that, for those $k$, almost all Gallai $k$-colorings use only two colors. 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Benevides, Jie Han, Josefran de Oliveira Bastos","submitted_at":"2018-12-26T18:59:12Z","abstract_excerpt":"An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$ with $k\\le 2^{n/4300}$, the number of Gallai colorings of $K_n$ that use at most $k$ given colors is $(\\binom{k}{2}+o_n(1))\\,2^{\\binom{n}{2}}$. Our result is asymptotically best possible and implies that, for those $k$, almost all Gallai $k$-colorings use only two colors. 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