Improved bounds on maximum sets of letters in sequences with forbidden alternations

Let $A_{s,k}(m)$ be the maximum number of distinct letters in any sequence which can be partitioned into $m$ contiguous blocks of pairwise distinct letters, has at least $k$ occurrences of every letter, and has no subsequence forming an alternation of length $s$. Nivasch (2010) proved that $A_{5, 2d+1}(m) = \theta( m \alpha_{d}(m))$ for all fixed $d \geq 2$. We show that $A_{s+1, s}(m) = \binom{m- \lceil \frac{s}{2} \rceil}{\lfloor \frac{s}{2} \rfloor}$ for all $s \geq 2$, $A_{5, 6}(m) = \theta(m \log \log m)$, and $A_{5, 2d+2}(m) = \theta(m \alpha_{d}(m))$ for all fixed $d \geq 3$.