Transitive Edge Coloring of Graphs and Dimension of Lattices

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,klE(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.

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Correspondence to András Gyárfás*.

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* Research supported in part by OTKA Grant T029074.

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Gyárfás*, A. Transitive Edge Coloring of Graphs and Dimension of Lattices. Combinatorica 22, 479–496 (2002).

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AMS Subject Classification (2000)

  • 05C15