Three-color Ramsey numbers for paths

We prove—for sufficiently large n—the following conjecture of Faudree and Schelp: $$R{\left( {P_{n} ,P_{n} ,P_{n} } \right)} = \left\{ {\begin{array}{*{20}c}{{2n - 1{\kern 1pt} \;{\text{for}}\;{\text{odd}}\;n,}} \\ {{{\text{2n - 2}}\;{\text{for}}\;{\text{even}}\;n,}} \\ \end{array} } \right.$$, for the three-color Ramsey numbers of paths on n vertices.

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