A_k(n;q,p) has tower height exactly ceil((k-1)/p) for fixed p and large q.
Loh,Directed paths: from Ramsey to Ruzsa and Szemer´ edi, (2015),arXiv:1505
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Tower heights for color-avoiding Ramsey numbers of monotone paths
A_k(n;q,p) has tower height exactly ceil((k-1)/p) for fixed p and large q.