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Forbidden configurations and Steiner designs

Abstract

Let \({\mathcal{F}}\) be a (0, 1) matrix. A (0, 1) matrix \({\mathcal{M}}\) is said to have \({\mathcal{F}}\) as a configuration if there is a submatrix of \({\mathcal{M}}\) which is a row and column permutation of \({\mathcal{F}}\). We say that a matrix \({\mathcal{M}}\) is simple if it has no repeated columns. For a given \({v \in \mathbb{N}}\), we shall denote by forb\({(v, \mathcal{F})}\) the maximum number of columns in a simple (0, 1) matrix with v rows for which \({\mathcal{F}}\) does not occur as a configuration. We say that a matrix \({\mathcal{M}}\) is maximal for \({\mathcal{F}}\) if \({\mathcal{M}}\) has forb\({(v, \mathcal{F})}\) columns. In this paper we show that for certain natural choices of \({\mathcal{F}}\), forb\({(v, \mathcal{F})\leq\frac{\binom{v}{t}}{t+1}}\). In particular this gives an extremal characterization for Steiner t-designs as maximal (0, 1) matrices in terms of certain forbidden configurations.

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Correspondence to Niranjan Balachandran.

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This is one of several papers published together in Designs, Codes and Cryptography on the special topic: “Combinatorics – A Special Issue Dedicated to the 65th Birthday of Richard Wilson”.

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Balachandran, N. Forbidden configurations and Steiner designs. Des. Codes Cryptogr. 65, 353–364 (2012). https://doi.org/10.1007/s10623-012-9699-x

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  • DOI: https://doi.org/10.1007/s10623-012-9699-x

Keywords

  • Forbidden configurations
  • Steiner designs
  • Nonuniform Ray–Chaudhuri–Wilson theorem

Mathematics Subject Classification

  • 05B30
  • 05D05