We present new short proofs to both the exact and the stability result of two extremal problems. The first result is about the extension of Turán’s theorem to hypergraphs, and the second result is about cancellative hypergraphs. Our proofs are concise and straightforward, but give a sharper version of stability theorems to both problems.

中文翻译：

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一些稳定性定理的新短证明

我们为两个极端问题的精确性和稳定性结果提供了新的简短证明。第一个结果是图兰定理扩展到超图，第二个结果是取消超图。我们的证明简洁明了，但是针对这两个问题给出了稳定性定理的更清晰版本。