Formations and generalized Davenport-Schinzel sequences

Abstract: Let $up(r, t) = (a_1 a_2 \dots a_r)<sup>t$. We investigate the problem of determining the maximum possible integer $n(r, t)$ for which there exist $2t-1$ permutations $\pi_1, \pi_2, \dots, \pi_{2t-1}$ of $1, 2, \dots, n(r, t)$ such that the concatenated sequence $\pi_1 \pi_2 \dots \pi_{2t-1}$ has no subsequence isomorphic to $up(r,t)$. This quantity has been used to obtain an upper bound on the maximum number of edges in $k$-quasiplanar graphs. It was proved by (Geneson, Prasad, and Tidor, Electronic Journal of Combinatorics, 2014) that $n(r, t) \le (r-1)<sup>{2<sup>{2t-2}}$. We prove that $n(r,t) = \Theta(r^{2t-1 \choose t})$, where the constant in the bound depends only on $t$. Using our upper bound in the case $t = 2$, we also sharpen an upper bound of (Klazar, Integers, 2002), who proved that $Ex(up(r,2),n) &lt; (2n+1)L$ where $L = Ex(up(r,2),K-1)+1$, $K = (r-1)⁴ + 1$, and $Ex(u, n)$ denotes the extremal function for forbidden generalized Davenport-Schinzel sequences. We prove that $K = (r-1)⁴ + 1$ in Klazar's bound can be replaced with $K = (r-1) \binom{r}{2}+1$. We also prove a conjecture from (Geneson, Prasad, and Tidor, Electronic Journal of Combinatorics, 2014) by showing for $t \geq 1$ that $$Ex(a b c (a c b)^{t} a b c, n) = n 2^{{\frac{1}{t!}\alpha(n)^{t}} \pm O(\alpha(n)<sup>{t-1})}$$. In addition, we prove that $$Ex(a b c a c b (a b c)^{t} a c b, n) = n 2^{{\frac{1}{(t+1)!}\alpha(n)^{t+1}} \pm O(\alpha(n)<sup>{t})}$$ for all $t \geq 1$.