Asymptotic behavior for joint Turán number for consecutive cycles
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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:
Erdös and Simonovits [1] proved that \( t(n,\{C_4,C_5\}) = \frac{1}{2\sqrt 2} n^{3/2}+O(n)\).
A problem on the Turán number for \( C_{2k1} \) and \(C_{2k}\) (proposed by Erdös and Simonovits [1])
Is it true that \[ t(n,\{C_{2k1},C_{2k}\}) = (1+o(1)) (\frac n 2)^{1+1/k}? \]
Bibliography  

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