Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two
cycles have the same length. P. Erdos raised the problem of
determining $f(n)$ (see J.A. Bondy and U.S.R. Murty, Graph Theory
with Applications (Macmillan, New York, 1976), p.247, Problem 11).
We present the problems,
conjectures related to this problems and we summarize the know results. We make the following conjecture:
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\noindent{\bf Conjecture } $$\lim\sb {n \to \infty} {f(n)-n \over \sqrt n} = \sqrt {2 + \frac{4}{9}}.$$