Constructing sparse Davenport–Schinzel sequences

Abstract

For any sequence u, the extremal function Ex(u,j,n) is the maximum possible length of a j-sparse sequence with n distinct letters that avoids u. We prove that if u is an alternating sequence abab… of length s, then Ex(u,j,n)=Θ(sn2) for all j≥2 and s≥n, answering a question of Wellman and Pettie (2018) and extending the result of Roselle and Stanton that Ex(u,2,n)=Θ(sn2) for any alternation u of length s≥n (Roselle and Stanton, 1971).Wellman and Pettie also asked how large must s(n) be for there to exist n-block DS(n,s(n)) sequences of length Ω(n2−o(1)). We answer this question by showing that the maximum possible length of an n-block DS(n,s(n)) sequence is Ω(n2−o(1)) if and only if s(n)=Ω(n1−o(1)). We also show related results for extremal functions of forbidden 0–1 matrices with any constant number of rows and extremal functions of forbidden sequences with any constant number of distinct letters.