Bounding extremal functions of forbidden $0-1$ matrices using $(r,s)$-formations

First, we prove tight bounds of $n 2^{\frac{1}{(t-2)!}α(n)^{t-2} \pm O(α(n)^{t-3})}$ 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.

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