Transitive edge coloring of graphs and dimension of lattices

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2 Citations (Scopus)


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 F1, F2,..., Fn 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 Fi ∩ Fj = Fk ∩ Fl. 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 languageEnglish
Pages (from-to)479-496
Number of pages18
Issue number4
Publication statusPublished - Dec 1 2002

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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