For every k≥4 and C>1 there is a linear k-uniform hypergraph H with off-diagonal Ramsey number r(H,K_n^{(k)}) at least the (k-2)-fold tower of 2^{(log n)^C}.
Erdős, Graph theory and probability
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Off-Diagonal Ramsey Numbers for Linear Hypergraphs
For every k≥4 and C>1 there is a linear k-uniform hypergraph H with off-diagonal Ramsey number r(H,K_n^{(k)}) at least the (k-2)-fold tower of 2^{(log n)^C}.