Number of pentagons in a triangle free graph

Problem [1]

Is it true that a triangle-free graph on \(5n\) vertices can contain at most \(n^5\) pentagons?

Györi \( ^{[2]}\) proved that such graph can have at most \( 3^3 5^4 / 2^{14}~ n^5 \approx 1.03 n^5\) triangles.


Bibliography
1 P. Erdös, On some problems in graph theory, combinatorial analysis and combinatorial number theory, Graph theory and combinatorics (Cambridge, 1983), 1-17, Academic Press, London-New York, 1984.

2 E. Györi, On the number of \( C_5\)'s in a triangle-free graph, Combinatorica 9 (1989), 101-102.