Lower Bounds on Davenport-Schinzel Sequences via Rectangular Zarankiewicz Matrices

An order-s Davenport-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 [1], Nivasch [12] and Pettie [15]) 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 0-1 matrices 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. Our lower bounds on λ(s, n) exhibit three qualitatively different behaviors depending on the size of s relative to n. When s ≤ log log n we show that λ(s, n)/n ≥ 2s grows exponentially with s. When s = no(1) we show λ(s, n)/n ≥ ( s 2 log logs n )log logs n 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.