C4-free subgraphs with large average degree

Motivated by a longstanding conjecture of Thomassen, we study how large the average degree graph needs to be imply that it contains C4-free subgraph with at least t. Kühn and Osthus showed an bound which is double exponential in t sufficient. We give short proof this bound, before reducing single exponential. That is, show any G $$2^{ct^{2}\log\ t}$$ (for some constant c > 0) Finally, construction improves lower for problem, showing initial must t3?o(1).