On the maximum lengths of Davenport-Schinzel sequences

The quantity N5(n) is the maximum length of a finite sequence over n symbols which has no two identical consecutive elements and no 5-term alternating subsequence. Improving the constant factor in the previous bounds of Hart and Sharir, and of Sharir and Agarwal, we prove that N5(n) < 2nα(n) +O(nα(n) ), where α(n) is the inverse to the Ackermann function. Quantities Ns(n) can be generalized and any finite sequence, not just an alternating one, can be assigned extremal function. We present a sequence with no 5-term alternating subsequence and with an extremal function n2.