Turán number for families of graphs behave like Turán number for one of the members
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For a finite family \( {\mathcal F}\) of graphs, let \( t(n, {\mathcal{F}})\) denote the smallest integer \( m\) that every graph on \( n\) vertices and \( m\) edges must contain a member of \( \mathcal{F}\) as a subgraph. Erdös and Simonovits [1] raised the following question:
A problem on the Turán number for families of graphs}
Prove that \[ t(n,{\mathcal F}) = O(t(n,G)) \] for some graph \(G\) in \(\mathcal{F}\).
Bibliography  

1 
P. Erdös and M. Simonovits, Compactness results in extremal graph
theory, Combinatorica 2 (1982), 275288.
