New Short Proofs to Some Stability Theorems
Abstract
We present new short proofs to both the exact and the stability results of two extremal problems. The first one is the extension of Tur\'an's theorem in hypergraphs, which was firstly studied by Mubayi MU06. The second one is about the cancellative hypergraphs, which was firstly studied by Bollob\'as BO74 and later by Keevash and Mubayi KM04. Our proofs are concise and straightforward, but give a sharper version of stability theorems to both problems.
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