Almost linear upper bounds on the length of general davenport-schinzel sequences

Davenport-Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We obtain almost linear upper bounds on the length λ_s(n) of Davenport-Schinzel sequences composed of n symbols in which no alternating subsequence is of length greater than s+1. These bounds are of the form O(nα(n)^O(α(n)5-3)), and they generalize and extend the tight bound Θ(nα(n)) obtained by Hart and Sharir for the special case s=3 (α(n) is the functional inverse of Ackermann's function), and also improve the upper bound O(n log*n) due to Szemerédi.