First, we prove tight bounds of $n 2^{ \frac{ 1}{ (t-2)!}\alpha(n)^{ t-2} \pm O(\alpha(n)^{ t-3})}$ on the extremal function of the forbidden pair of ordered sequences $(1 2 3 \ldots k)^t$ and $(k \ldots 3 2 1)^t$ using bounds on a class of sequences called $(r,s)$-formations. Then, we show how an analogous method can be used to derive similar bounds on the extremal functions of forbidden pairs of $0-1$ matrices consisting of horizontal concatenations of identical identity matrices and their horizontal reflections.
Categories: math.CO cs.DM