**Conjecture**For all positive integers and , there exists an integer such that every graph of average degree at least contains a subgraph of average degree at least and girth greater than .

This conjecture is true for regular graphs as observed by Alon (see [KO]). The case was proved in [KO].

## Bibliography

[KO] D. Kühn and D. Osthus, Every graph of sufficiently large average degree contains a C4-free subgraph of large average degree, Combinatorica, 24 (2004), 155-162.

*[T] C. Thomassen, Girth in graphs, J. Combin. Theory B 35 (1983), 129–141.

* indicates original appearance(s) of problem.