Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, January 12, 2006, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Samuel Zürcher

On 0-1-matrices and small excluded submatrices

We consider 0/1 matrices and say that an n*n matrix A contains a k*l matrix P (the pattern) if and only if we can find a submatrix B of A (that is obtained by deleting rows and columns of A) so that (P)_i,j = 1 implies (B)_i,j = 1. Otherwise, we say that A avoids P. For a given pattern P, we are interested in the maximum number of 1 entries that an n*n matrix can have avoiding the pattern P; written as ex(n, P). This extremal function is known for all patterns with at most four 1 entries and each one of those patterns is member in one of five classes; we'll show examples for each of those classes and sketch some of the proofs for the bounds on ex(n, P). In a second part of the talk, we'll show reductions that can be used to determine ex(n, P') for a pattern P' with previously unknown complexity, based on the knowledge of the complexity of a somehow related pattern P.

This talk is mainly based on work by Tardos, Marcus, Füredi, Hajnal, and Keszegh.

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