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Motivated by a longstanding conjecture of Thomassen, we study how large theaverage degree of a graph needs to be to imply that it contains a $C_4$-freesubgraph with average degree at least $t$. K\"uhn and Osthus showed that anaverage degree bound which is double exponential in t is sufficient. We give ashort proof of this bound, before reducing it to a single exponential. That is,we show that any graph $G$ with average degree at least $2^{ct^2\log t}$ (forsome constant $c>0$) contains a $C_4$-free subgraph with average degree atleast $t$. Finally, we give a construction which improves the lower bound forthis problem, showing that this initial average degree must be at least$t^{3-o(1)}$.

C4-free subgraphs with large average degree