Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erdős–Gallai Theorem in random graphs. In particular, we determine, up to a constant factor, the maximum number of edges in a $P_n$-free subgraph of $G(N,p)$, practically for all values of $N,n$ and $p$. Our work is also motivated by the recent progress on the size-Ramsey number of paths.

中文翻译：

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随机图的Erdős-Gallai定理的类似物

最近，在随机环境中已经证明了许多经典极值定理的变体。我们对现有结果进行补充，在随机图中扩展了Erdős-Gallai定理。尤其是，我们确定一个恒定因子，即$P_{ñ}$的免费子图 $G（ñ，p）$，实际上适用于 $ñ，ñ$ 和 $p$。我们的工作也受到最近在规模-拉姆西路径上取得进展的推动。