On a problem of Erdos and Rothschild on edges in triangles

Abstract

Erdos and Rothschild asked to estimate the maximum number, denoted by h(n; c), such that every n-vertex graph with at least cn2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least h(n; c) triangles. In particular, Erdos asked in 1987 to determine whether for every c > 0 there is epsilon > 0 such that h(n; c) > n epsilon for all su ciently large n. We prove that h(n; c) = nO(1= log log n) for every xed c < 1=4. This gives a negative answer to the question of Erdos, and is best possible in terms of the range for c, as it is known that every n-vertex graph with more than n2=4 edges contains an edge that is in at least n=6 triangles.