Sparse sets in the complements of graphs with given girth

A set of edges in a graph is sparse if no two of these edges belong to the same clique. It is shown here that if a graph has girth at least 5, 6 or 8 then the largest number of edges in a sparse set in its complement is at most 8, 7 or 6, respectively; this result is complete and best possible. It follows that if ¿ 0, then for su1ciently large n there exists a graph with n vertices and chromatic number greater than n1=3− , n1=4− or n1=6− whose complement contains no sparse set with more than 8, 7 or 6 edges, respectively. c © 2001 Elsevier Science B.V. All rights reserved.