In 1975, P. Erd\H os proposed the problem of determining the
maximum number $f(n)$ of edges in a graph on $n$ vertices in
which any two cycles are of different
lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple graph on $n$ vertices
in which any two cycles are of different
lengths. Let $M_n$ be the set of simple graphs on $n$ vertices in which any two cycles are of different
lengths and with the edges of $f^{\ast}(n)$. Let $mc(n)$ be the maximum cycle length for all $G \in M_n$.
In this paper, it is proved that for $n$ sufficiently large, $mc(n)\leq \frac{15}{16}n.$
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We make the following conjecture:
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\noindent{\bf Conjecture.} $$\lim_{n \rightarrow \infty} {mc(n)\over n}= 0.$$