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

Let As,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 A5,2d+1(m) = θ(mαd(m)) for all fixed d ≥ 2. We show that As+1,s(m) = ( m−⌈ s 2 ⌉ ⌊ s 2 ⌋ ) for all s ≥ 2, A5,6(m) = θ(m log logm), and A5,2d+2(m) = θ(mαd(m)) for all fixed d ≥ 3.