New Short Proofs to Some Stability Theorems

TL;DR

This paper introduces concise proofs for stability theorems in extremal hypergraph problems, improving existing results on Turán's theorem extension and cancellative hypergraphs.

Contribution

It provides new, shorter, and sharper proofs for stability results in hypergraph extremal problems, extending prior work by Mubayi, Bollobás, Keevash, and Mubayi.

Findings

Sharper stability theorems for hypergraph Turán problems

Concise proofs that simplify existing complex arguments

Improved bounds in extremal hypergraph configurations

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