Three-Color Ramsey Numbers For Paths

  title={Three-Color Ramsey Numbers For Paths},
  author={Andr{\'a}s Gy{\'a}rf{\'a}s and Mikl{\'o}s Ruszink{\'o} and G{\'a}bor N. S{\'a}rk{\"o}zy and Endre Szemer{\'e}di},
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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