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

Abstract: First, we prove tight bounds of $$n 2^{{\frac{1}{(t-2)!}\alpha(n)^{t-2}} \pm O(\alpha(n)<sup>{t-3})}$$ on the extremal function of the forbidden pair of ordered sequences $(1 2 3 \ldots k)<sup>t$ and $(k \ldots 3 2 1)<sup>t$ 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.