A stepping-up lemma for monotone paths with bounded color complexity

For positive integers $n, k, q, p$, let $A_k(n; q, p)$ be the largest integer $N$ such that there exists an edge coloring of $K_N^{(k)}$ with $q$ colors that does not contain a tight monotone path of length $n$ that consists of at most $p$ colors. In the case $p = 1$, this coincides with the ordinary Ramsey number of a tight monotone path, and it is known that $A_k(n; q, 1) = T_{k-2}(n^{\Theta(q)})$, proved by Moshkovitz and Shapira. Recently, Mulrenin, Pohoata, and Zakharov showed that whenever $p > \frac{q}{2}$, an improved upper bound $A_k(n; q, p) \leq T_{k-3}(n^{O(q)})$ holds, without any accompanying lower bounds. In this paper, we obtain the first non-trivial lower bound by developing a novel variant of the classical stepping-up lemma applicable to an Erd\H{o}s--Szekeres-type problem in which one seeks a tight monotone path spanning at most $p$ colors. In particular, we show that for any fixed $p \geq 1$, there exists a constant $C_p > 0$ that only depends on $p$ such that $$A_{k}(n; q, p) \geq T_{\lfloor k/ C_p \rfloor}\left(n^{\omega_q(1)}\right)$$ holds for all sufficiently large $n, k, q$ compared with $p$, that is, a tower function whose height grows linearly in $k$. A key ingredient in our proof is establishing a finite analogue of the celebrated Morse--Hedlund theorem, which may be of independent interest.