On the size of graphs with all cycle having distinct length

AbstractLet ƒ(n) be the maximum number of edges in a graph on n vertices in which no two cycles have the same length. In 1975, Erdős raised the question of determining ƒ(n) (see Bondy and Murty (1976)). In this note, we prove that for n⩾36·5t2−4·5t+1 one has ƒ(n)⩾n+9t−1. We conjecture that lim(ƒ(n)−n)/√n⩽√3