The r-colour size Ramsey number of P_k equals Theta((r^2 log r) k) for fixed r >= 2 and k >= 100 log r.
Long cycles in locally expanding graphs, with applications.Combinatorica, 39(1):135–151, 2019
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The multicolour size Ramsey number of a path
The r-colour size Ramsey number of P_k equals Theta((r^2 log r) k) for fixed r >= 2 and k >= 100 log r.