For any fixed p, A_k(n; q, p) is at least a tower of height linear in k with exponent n to a slowly growing function of q.
Loh,Directed paths: from Ramsey to Ruzsa and Szemer´ edi, (2015),arXiv:1505
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A stepping-up lemma for monotone paths with bounded color complexity
For any fixed p, A_k(n; q, p) is at least a tower of height linear in k with exponent n to a slowly growing function of q.