Adding an edge to an extremal \(4\)-cycle-free graph gives two \(4\)-cycles

Conjecture (proposed by Erdös and Simonovits \( ^{[2]}\))
Every graph \( G \) on \(n\) vertices and \(t(n,C_4)+1\) edges contains at least two copies of \(C_4 \) when \(n\) is large.
Rademacher first observed \( ^{[1]}\) that every graph on \( n\) vertices and \( t(n,K_3)+1\) edges contains at least \( \lfloor n/2 \rfloor\) triangles. Similar questions can be asked for a general graph \( H\), but relatively few results are known for such problems (except for some trivial cases such as stars or disjoint edges \( ^{[2]}\)).

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 P. Erdös and M. Simonovits, Cube-supersaturated graphs and related problems, Progress in graph theory (Waterloo, Ont., 1982), 203-218, Academic Press, Toronto, 1984.