Extremal combinatorics of matrices, sequences and sets of permutations

Title: Extremal combinatorics of matrices, sequences and sets of permutations Author: Josef Cibulka Department: Department of Applied Mathematics Supervisor: Doc. RNDr. Pavel Valtr, Dr., Department of Applied Mathematics Abstract: This thesis studies questions from the areas of the extremal theory of {0, 1}-matrices, sequences and sets of permutations, which found many ap- plications in combinatorial and computational geometry. The VC-dimension of a set P of n-element permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. We show lower and upper bounds quasiexponential in n on the maximum size of a set of n-element permutations with VC-dimension bounded by a constant. This is used in a paper of Jan Kynčl to considerably improve the upper bound on the number of weak isomorphism classes of com- plete topological graphs on n vertices. For some, mostly permutation, matrices M, we give new bounds on the number of 1-entries an n × n M-avoiding matrix can have. For example, for every even k, we give a construction of a matrix with k2 n/2 1-entries that avoids one specific k-permutation matrix. We also give almost tight bounds on the maximum number of 1-entries in matrices avoiding a fixed layered..