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

Let c denote the largest constant such that every C ₆-free graph G contains a bipartite and C ₄-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 = 3/8. More generally, we show that for any ε > 0, and any integer k ≧ 2, there is a C _2k-free graph G ^' which does not contain a bipartite subgraph of girth greater than 2k with more than a fraction (1 − ₂₂ ¹ _{k −2} ⁾ _2k ² _{− 1} (1 + ε) of the edges of G ^'. There also exists a C _2k-free graph G ^'' which does not contain a bipartite and C ₄-free subgraph with more than a fraction (1 − _2k ¹ ₋₁ ⁾ _{k −} ¹ ₁ (1 + ε) 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 (1 − _ba ¹ ₋₁ ⁾ (1 + ε) 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 C ₄-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 ≧ 3).