Sharp bounds on Davenport-Schinzel sequences of every order

• Authors: Seth Pettie
• Venue: Proceedings of the twenty-ninth annual symposium on Computational geometry
• Year: 2013

One of the oldest unresolved problems in extremal combinatorics is to determine the maximum length of Davenport-Schinzel sequences, where an order-s DS sequence is defined to be one over an n-letter alphabet that avoids alternating subsequences of the form a ··· b ··· a ··· b ··· with length s+2. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since become an indispensable tool in computational geometry and the analysis of discrete geometric structures. Let DS{s}(n) be the extremal function for such sequences. What is DS{s} asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch) when s is even or s ≤ 3. However, since the work of Agarwal, Sharir, and Shor in the 1980s there has been a persistent gap in our understanding of the odd orders, a gap that is just as much qualitative as quantitative. In this paper we establish the following bounds on DS{s}(n) for every order s. DS{s}(n) = {n, s=1; 2n-1, s=2; 2nα(n) + O(n), s=3; Θ(n2α(n)), s=4; Θ(nα(n)2α(n)), s=5; n2(1 + o(1))αt(n)/t!, s ≥ 6, ; t = ⌊(s-2)/2⌋ These results refute a conjecture of Alon, Kaplan, Nivasch, Sharir, and Smorodinsky and run counter to common sense. When $s$ is odd, DS{s} behaves essentially like DS{s-1}.