On subgraphs of C _{2 k}-free graphs and a problem of Kühn and Osthus

Let c denote the largest constant such that every C₆-free graph G contains a bipartite and C₄-free subgraph having a fraction c of edges of G. Gy 'ri, Kensell and Tompkins showed that 3/8 (c)1/2 c (c)1/2 2/5. We prove that c = 38. More generally, we show that for any ϵ > 0, and any integer k (c)3/4 2, there is a C_2k-free graph <![CDATA[ $G'$ ]]> which does not contain a bipartite subgraph of girth greater than 2k with more than a fraction <![CDATA[ $$\Bigl(1-\frac{1}{2^{2k-2}}\Bigr)\frac{2}{2k-1}(1+\varepsilon)$$ ]]> of the edges of <![CDATA[ $G'$ ]]>. There also exists a C_2k-free graph <![CDATA[ $G''$ ]]> which does not contain a bipartite and C₄-free subgraph with more than a fraction <![CDATA[ $$\Bigl(1-\frac{1}{2^{k-1}}\Bigr)\frac{1}{k-1}(1+\varepsilon)$$ ]]> of the edges of <![CDATA[ $G''$ ]]>.One of our proofs uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erd 's. For any ϵ > 0, and any integers a, b, k (c)3/4 2, there exists an a-uniform hypergraph H of girth greater than k which does not contain any b-colourable subhypergraph with more than a fraction <![CDATA[ $$\Bigl(1-\frac{1}{b^{a-1}}\Bigr)(1+\varepsilon)$$ ]]> of the hyperedges of H. We also prove further generalizations of this theorem.In addition, we give a new and very short proof of a result of Kühn and Osthus, which states that every bipartite C_2k-free graph G contains a C₄-free subgraph with at least a fraction 1/(k-1) of the edges of G. We also answer a question of Kühn and Osthus about C_2k-free graphs obtained by pasting together C_2l's (with k >l (c)3/4 3).