Lower bounds on Davenport–Schinzel sequences via rectangular Zarankiewicz matrices

• Published in: Discrete mathematics Vol. 341; no. 7; pp. 1987 - 1993
• Main Authors: Wellman, Julian, Pettie, Seth
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.07.2018
• Subjects: Davenport–Schinzel problem; Forbidden submatrix; Forbidden subsequence; Zarankiewicz problem; Zarankiewicz problem; Forbidden submatrix; Davenport–Schinzel problem; Forbidden subsequence
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2018.03.023

An order-sDavenport–Schinzel sequence over an n-letter alphabet is one avoiding immediate repetitions and alternating subsequences with length s+2. The main problem is to determine the maximum length of such a sequence, as a function of n and s. When s is fixed this problem has been settled (see Agarwal, Sharir, and Shor, 1989, Nivasch, 2010 and Pettie, 2015) but when s is a function of n, very little is known about the extremal function λ(s,n) of such sequences. In this paper we give a new recursive construction of Davenport–Schinzel sequences that is based on dense 0–1matrices avoiding large all-1 submatrices (aka Zarankiewicz’s Problem ). In particular, we give a simple construction of n2∕t×n matrices containing n1+1∕t 1s that avoid t×2 all-1 submatrices. (This result seems to be absent from the literature on Zarankiewicz’s problem, but it may be considered folklore among experts in this area [Z. Füredi, personal communication, 2017].) Our lower bounds on λ(s,n) exhibit three qualitatively different behaviors depending on the size of s relative to n. When s≤loglogn we show that λ(s,n)∕n≥2s grows exponentially with s. When s=no(1) we show λ(s,n)∕n≥(s2loglogsn)loglogsn grows faster than any polynomial in s. Finally, when s=Ω(n1∕t(t−1)!), λ(s,n)=Ω(n2s∕(t−1)!) matches the trivial upper bound O(n2s) asymptotically, whenever t is constant.