Algebraic approach to stability results for Erd\H{o}s-Ko-Rado theorem

Celebrated results often unfold like episodes in a long-running series. In the field of extremal set thoery, Erd\H{o}s, Ko, and Rado in 1961 established that any $k$-uniform intersecting family on $[n]$ has a maximum size of $\binom{n-1}{k-1}$, with the unique extremal structure being a star. In 1967, Hilton and Milner followed up with a pivotal result, showing that if such a family is not a star, its size is at most $\binom{n-1}{k-1} - \binom{n-k-1}{k-1} + 1$, and they identified the corresponding extremal structures. In recent years, Han and Kohayakawa, Kostochka and Mubayi, and Huang and Peng have provided the second and third levels of stability results in this line of research. In this paper, we provide a unified approach to proving the stability result for the Erd\H{o}s-Ko-Rado theorem at any level. Our framework primarily relies on a robust linear algebra method, which leverages appropriate non-shadows to effectively handle the structural complexities of these intersecting families.