### 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 |
---|---|

Pages (from-to) | 280-290 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 76 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 1999 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series B*,

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

**Norm-Graphs : Variations and Applications.** / Alon, Noga; Rónyai, L.; Szabó, Tibor.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 76, no. 2, pp. 280-290. https://doi.org/10.1006/jctb.1999.1906

}

TY - JOUR

T1 - Norm-Graphs

T2 - Variations and Applications

AU - Alon, Noga

AU - Rónyai, L.

AU - Szabó, Tibor

PY - 1999/7

Y1 - 1999/7

N2 - 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 12n5/3 edges, containing no copy of K3, 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 K3, 3 is (1+o(1))k3. 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(mt) and whose discrepancy is Ω(n1/2-1/2tlogn). This settles a problem of Matoušek.

AB - 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 12n5/3 edges, containing no copy of K3, 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 K3, 3 is (1+o(1))k3. 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(mt) and whose discrepancy is Ω(n1/2-1/2tlogn). This settles a problem of Matoušek.

UR - http://www.scopus.com/inward/record.url?scp=0001100795&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001100795&partnerID=8YFLogxK

U2 - 10.1006/jctb.1999.1906

DO - 10.1006/jctb.1999.1906

M3 - Article

AN - SCOPUS:0001100795

VL - 76

SP - 280

EP - 290

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -