New Short Proofs to Some Stability Theorems

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'{a}n’s theorem in hypergraphs, which was firstly studied by Mubayi $\text{cite}MU06$. The second one is about the cancellative hypergraphs, which was firstly studied by Bollob'{a}s $\text{cite}BO74$ and later by Keevash and Mubayi $\text{cite}KM04$. Our proofs are concise and straightforward, but give a sharper version of stability theorems to both problems.