Bounding Sequence Extremal Functions with Formations

An (r, s)-formation is a concatenation of s permutations of r letters. If u is a sequence with r distinct letters, then let Ex (u, n) be the maximum length of any r-sparse sequence with n distinct letters which has no subsequence isomorphic to u. For every sequence u define fw(u), the formation width of u, to be the minimum s for which there exists r such that there is a subsequence isomorphic to u in every (r, s)-formation. We use fw(u) to prove upper bounds on Ex (u, n) for sequences u such that u contains an alternation with the same formation width as u. We generalize Nivasch’s bounds on Ex ((ab)t, n) by showing that fw((12 . . . l)t) = 2t − 1 and Ex ((12 . . . l)t, n) = n2 1 (t−2)!α(n) t−2±O(α(n)t−3) for every l > 2 and t > 3, such that α(n) denotes the inverse Ackermann function. Upper bounds on Ex ((12 . . . l)t, n) have been used in other papers to bound the maximum number of edges in k-quasiplanar graphs on n vertices with no pair of edges intersecting in more than O(1) points. If u is any sequence of the form avav′a such that a is a letter, v is a nonempty sequence excluding a with no repeated letters and v′ is obtained from v by only moving the first letter of v to another place in v, then we show that fw(u) = 4 and Ex (u, n) = Θ(nα(n)). Furthermore we prove that fw(abc(acb)t) = 2t + 1 and Ex (abc(acb)t, n) = n2 1 (t−1)!α(n) t−1±O(α(n)t−2) for every t > 2.