I need a reference on any upper bound on $R(n, n, \dots, n)$ with $k$ arguments. For example, the standard recurrent bound gives something like $k^{kn}$, but I cannot find any written explicit bound.
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1$\begingroup$ You can get $k^{kn}$ by restricting to the largest coloured neighbourhood of an arbitrary vertex $kn$ times, then doing a second pass to get $n$ vertices where all the forward edges are the same colour. $\endgroup$– Ben BarberCommented Jun 2, 2017 at 11:00
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