For each fixed k, the canonical Ramsey number ER(K^(k)_{t,...,t}) is at most t^{t^{k^2}} for large t, giving a single-exponential upper bound.
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Canonical Ramsey numbers for partite hypergraphs
For each fixed k, the canonical Ramsey number ER(K^(k)_{t,...,t}) is at most t^{t^{k^2}} for large t, giving a single-exponential upper bound.