An old problem of Erdos: A graph without two cycles of the same length

In 1975, P. Erdos 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 *(n) be the maximum number of edges in a simple graph on n vertices in which any two cycles are of different lengths. Let Mn be the set of simple graphs on n vertices and f *(n) edges in which any two cycles are of different lengths. Let mc(n) be the maximum cycle length for all G E Mn. In this paper, it is proved that for n sufficiently large, mc(n) <= 15 16 n. We make the following conjecture: Conjecture. lim n ->infinity mc(n) n = 0. (c) 2023 Elsevier B.V. All rights reserved.