A conjecture of Erdős from 1967 asserts that any graph on n vertices which does not contain a fixed r-degenerate bipartite graph F has at most $C{n}^{2-1/r}$ edges, where C is a constant depending only on F. We show that this bound holds for a large family of r-degenerate bipartite graphs, including all r-degenerate blow-ups of trees. Our results generalise many previously proven cases of the Erdős conjecture, including the related results of Füredi and Alon, Krivelevich and Sudakov. Our proof uses supersaturation and a random walk on an auxiliary graph.

中文翻译：

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图兰树木爆炸次数

Erdős 在 1967 年的一个猜想断言，任何在n个顶点上的不包含固定r -退化二部图F的图至多有$C{n}^{2-1/r}$边，其中C是仅取决于F的常数。我们证明了这个界限适用于一大群r简并二部图，包括所有r简并树的爆炸。我们的结果概括了许多先前已证实的 Erdős 猜想案例，包括 Füredi 和 Alon、Krivelevich 和 Sudakov 的相关结果。我们的证明在辅助图上使用过饱和和随机游走。