### Abstract

We explore properties of edge colorings of graphs defined by set intersections. An edge coloring of a graph G with vertex set V = {1, 2,..., n} is called transitive if one can associate sets F_{1}, F_{2},..., F_{n} to vertices of G so that for any two edges ij, kl ∈ E(G), the color of ij and kl is the same if and only if F_{i} ∩ F_{j} = F_{k} ∩ F_{l}. The term transitive refers to a natural partial order on the color set of these colorings. We prove a canonical Ramsey type result for transitive colorings of complete graphs which is equivalent to a stronger form of a conjecture of A. Sali on hypergraphs. This - through the reduction of Sali - shows that the dimension of n-element lattices is o(n) as conjectured by Füredi and Kahn. The proof relies on concepts and results which seem to have independent interest. One of them is a generalization of the induced matching lemma of Ruzsa and Szemerédi for transitive colorings.

Original language | English |
---|---|

Pages (from-to) | 479-496 |

Number of pages | 18 |

Journal | Combinatorica |

Volume | 22 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1 2002 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics