Tightish Bounds on Davenport-Schinzel Sequences

Details Tightish Bounds on Davenport-Schinzel Sequences Tightish Bounds on Davenport-Schinzel Sequences

2012-04-04

arxiv.org/abs/1204.1086v1

Computer Science

Discrete Mathematics

Scientific paper

Let DS_s(n) be the extremal function of order-s Davenport-Schinzel sequences over an n-letter alphabet. Together with existing bounds due to Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch, we give the following essentially tight bounds on DS_s(n) for all s: DS_s(n) = n, for s=1, 2n-1, for s=2, Theta(n alpha(n)), for s=3, Theta(n 2^{alpha(n)}), for s=4, Omega(n alpha(n) 2^{alpha(n)}) and O(n alpha^2(n) 2^{alpha(n)}), for s=5, n 2^{(1 \pm o(1)) alpha^t(n)/t!}, for s\geq6 and t = floor{(s-2)/2} These bounds refute conjectures due to Alon et al., Nivasch, and Pettie.

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