### Abstract

We describe several variants of the norm-graphs introduced by Kollár, Rónyai, and Szabó and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 12n^{5/3} edges, containing no copy of K_{3, 3}, thus slightly improving an old construction of Brown. We also prove that the maximum number of vertices in a complete graph whose edges can be colored by k colors with no monochromatic copy of K_{3, 3} is (1+o(1))k^{3}. This answers a question of Chung and Graham. In addition we prove that for every fixed t, there is a family of subsets of an n element set whose so-called dual shatter function is O(m^{t}) and whose discrepancy is Ω(n^{1/2-1/2t}logn). This settles a problem of Matoušek.

Original language | English (US) |
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Pages (from-to) | 280-290 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 76 |

Issue number | 2 |

DOIs | |

State | Published - Jul 1 1999 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*76*(2), 280-290. https://doi.org/10.1006/jctb.1999.1906