C6-free bipartite graphs and product representation of squares

  title={C6-free bipartite graphs and product representation of squares},
  author={Ervin Gy{\"o}ri},
  journal={Discrete Mathematics},
for any F > 0 if n is large enough (depending on k and E). The case k = 2 is nearly trivial since Fz(n) is equal to the number of square-free integers not exceeding n which is about (6/7c2)n. For the case of even integers k > 2, they worked out a number-theoretic method which reduces the problem to the study of Ck-free unbalanced bipartite graphs. For k = 4, the corresponding graph-theoretic problem can be settled relatively easily so that they got the following quite satisfactory estimates: 

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