Forbidden formations in 0-1 matrices
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Keszegh (2009) proved that the extremal function ex(n, P) of any forbidden light 2-dimensional 0-1 matrix P is at most quasilinear in n, using a reduction to generalized Davenport-Schinzel sequences. We extend this result to multidimensional matrices by proving that any light d-dimensional 0-1 matrix P has extremal function ex(n, P,d) = O(n^d-12^α(n)^t) for some constant t that depends on P. To prove this result, we introduce a new family of patterns called (P, s)-formations, which are a generalization of (r, s)-formations, and we prove upper bounds on their extremal functions. In many cases, including permutation matrices P with at least two ones, we are able to show that our (P, s)-formation upper bounds are tight.
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