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,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.