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. Győri, Kensell and Tompkins showed that 3/8 ⩽ c ⩽ 2/5. We prove that c = 38. More generally, we show that for any ε > 0, and any integer k ⩾ 2, there is a C2k-free graph $G'$ which does not contain a bipartite subgraph of girth greater than 2k with more than a fraction $$\Bigl(1-\frac{1}{2^{2k-2}}\Bigr)\frac{2}{2k-1}(1+\varepsilon)$$ of the edges of $G'$ . There also exists a C2k-free graph $G''$ which does not contain a bipartite and C4-free subgraph with more than a fraction $$\Bigl(1-\frac{1}{2^{k-1}}\Bigr)\frac{1}{k-1}(1+\varepsilon)$$ of the edges of $G''$ .One of our proofs uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erdős. For any ε > 0, and any integers a, b, k ⩾ 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 $$\Bigl(1-\frac{1}{b^{a-1}}\Bigr)(1+\varepsilon)$$ 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 C2k-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 C2k-free graphs obtained by pasting together C2l’s (with k >l ⩾ 3).

中文翻译：

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关于无 C2k 图的子图和 Kühn 和 Osthus 的问题

让C表示最大常数，使得每个C6- 自由图G包含一个二分和C4- 有分数的自由子图C的边缘G. Győri、Kensell 和 Tompkins 表明 3/8 ⩽C⩽ 2/5。我们证明C= 38. 更一般地，我们证明对于任何ε> 0，以及任何整数ķ⩾ 2，有一个C2ķ- 自由图$G'$不包含周长大于 2 的二分子图ķ超过一小部分$$\Bigl(1-\frac{1}{2^{2k-2}}\Bigr)\frac{2}{2k-1}(1+\varepsilon)$$的边缘$G'$. 还存在一个C2ķ- 自由图$G''$其中不包含二分和C4- 超过分数的自由子图$$\Bigl(1-\frac{1}{2^{k-1}}\Bigr)\frac{1}{k-1}(1+\varepsilon)$$的边缘$G''$. 我们的一个证明使用以下陈述，我们使用概率思想证明，推广 Erdős 的定理。对于任何ε> 0，以及任何整数一种,b,ķ⩾ 2，存在一个一种- 均匀超图H周长大于ķ其中不包含任何b- 可着色的子超图，超过一个分数$$\Bigl(1-\frac{1}{b^{a-1}}\Bigr)(1+\varepsilon)$$的超边H. 我们还证明了这个定理的进一步推广。此外，我们对 Kühn 和 Osthus 的结果给出了一个新的非常简短的证明，它指出每个二分C2ķ- 自由图G包含一个C4- 至少有分数 1/(ķ-1) 的边缘G. 我们还回答了 Kühn 和 Osthus 关于C2ķ- 通过粘贴在一起获得的自由图C2l的（与ķ>l⩾ 3)。