Improved bounds and new techniques for Davenport - Schinzel sequences and their generalizations

We present several new results regarding λ_s(n), the maximum length of a Davenport - Schinzel sequence of order s on n distinct symbols. First, we prove that λ_s(n) ≤n·²(1/t!)α(n)^t + O(α(n) ^t-1) for s≥4 even, and λ_s(n)·n ²(1/t)α(n)^t log₂ α(n) + O(α(n)^t) for s≥3 odd, where t = ⌊(s-2)/2⌋, and α(n) denotes the inverse Ackermann function. The previous upper bounds, by Agarwal et al. [1989], had a leading coefficient of 1 instead of 1/t! in the exponent. The bounds for even s are now tight up to lower-order terms in the exponent. These new bounds result from a small improvement on the technique of Agarwal et al. More importantly, we also present a new technique for deriving upper bounds for λ_s(n). This new technique is very similar to the one we applied to the problem of stabbing interval chains [Alon et al. 2008]. With this new technique we: (1) re-derive the upper bound of λ₃(n)≤2n α(n) + O(n√α(n)) (first shown by Klazar [1999]); (2) re-derive our own new upper bounds for general s and (3) obtain improved upper bounds for the generalized Davenport - Schinzel sequences considered by Adamec et al. [1992]. Regarding lower bounds, we show that λ₃(n)≥2n α(n) - O(n) (the previous lower bound (Sharir and Agarwal, 1995) had a coefficient of 1/2), so the coefficient 2 is tight. We also present a simpler variant of the construction of Agarwal et al. [1989] that achieves the known lower bounds of λ_s(n) ≥n·²(1/t!) α(n)^t - O(α(n) ^t-1) for s≥4 even.