We present two extensions of the linear bound, due to Marcus and Tardos, on the number of 1's in an n × n 0-1 matrix avoiding a fixed permutation matrix. We first extend the linear bound to hypergraphs with ordered vertex sets and, using previous results of Klazar, we prove an exponential bound on the number of hypergraphs on n vertices which avoid a fixed permutation. This, in turn, solves various conjectures of Klazar as well as a conjecture of Brändén and Mansour. We then extend the original Füredi-Hajnal problem from ordinary matrices to d-dimensional matrices and show that the number of 1's in a d-dimensional 0-1 matrix with side length n which avoids a d-dimensional permutation matrix is O(n d−1 ).

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A class of permutations $\Pi$ is called closed if $\pi\subset\sigma\in\Pi$ implies $\pi\in\Pi$, where the relation $\subset$ is the natural containment of permutations. Let $\Pi_n$ be the set of all permutations of $1,2,\dots,n$ belonging to $\Pi$. We investigate the counting functions $n\mapsto|\Pi_n|$ of closed classes. Our main result says that if $|\Pi_n| < 2^{n-1}$ for at least one $n\ge 1$, then there is a unique $k\ge 1$ such that $F_{n,k}\le |\Pi_n|\le F_{n,k}\cdot n^c$ holds for all $n\ge 1$ with a constant $c>0$. Here $F_{n,k}$ are the generalized Fibonacci numbers which grow like powers of the largest positive root of $x^k-x^{k-1}-\cdots-1$. We characterize also the constant and the polynomial growth of closed permutation classes and give two more results on these.

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The extremal function Ex(u,n) (introduced in the theory of Davenport-Schinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa.., the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following the idea of J. Ne~et~il) we generalize this concept for arbitrary sequence u. We summarize the already known properties of Ex(u,n) and we present also two new theorems which give good upper bounds on Ex(u,n) for u consisting of (two) smaller subsequences u i provided we have good upper bounds on Ex(ui,n ). We use these theorems to describe a wide class of sequences u ("linear sequences") for which Ex(u,n)= O(n). Both theorems are used for obtaining new superlinear upper bounds as well. We partially characterize linear sequences over three symbols. We also present several problems about Ex(u,n).