Lagrangian densities of 4-uniform matchings and degree stability of extremal hypergraphs

The Lagrangian density of an r-uniform graph F is ${\pi }_{\lambda }\left(F\right)=\sup \{r!\lambda (G):GisF\text{-}free\}$, where $\lambda \left(G\right)$ is the Lagrangian of an r-uniform graph G. Hypergraph Lagrangian has been a helpful tool in extremal combinatorics. Let $M_t^r$ denote the r-uniform matching with size t. The well-known Erdős Matching conjecture proposed that the Turán number of $M_t^r$ is $\max \left\{e\right(K_{tr-1}^r),e(S_{t-1}^r\left(n\right)\left)\right\}$, where $K_{tr-1}^r$ is the complete r-graph on $rt-1$ vertices and $S_{t-1}^r\left(n\right)$ is the r-graph with vertex set $\left[n\right]$ and with edge set $E\left(S_{t-1}^r\right(n\left)\right)=\{e\in \left(\begin{array}{c}\left[n\right]\\ r\end{array}\right):|e\cap [t-1]|\ge 1\}$. Regarding Lagrangian density of hypergraph matchings, Jiang, Peng and Wu [22] (Wu [34] as well) conjectured that the property similar to Erdős Matching Conjecture holds, precisely, they conjectured that ${\pi }_{\lambda }\left(M_t^r\right)=max\{r!\lambda (K_{tr-1}^r),r!{\lim }_{n\to \infty }\lambda (S_{t-1}^r\left)\right(n\left)\right\}$. Hefetz and Keevash [18] confirmed for $M_2^3$, Jiang, Peng and Wu [22] confirmed for $M_t^3$, Wu, Peng and Chen [35] confirmed for $M_2^4$, Bene Watts, Norin and Yepremyan [2] confirmed for $M_2^r$ for $r\ge 4$, Wu [34] confirmed for $M_3^4$. In this paper, we show that ${\pi }_{\lambda }\left(M_t^4\right)=4!\lambda \left(K_{4t-1}^4\right)$ if $t\ge 4$, this settles the conjecture for $M_t^4$. Our result has also interesting applications. Combining our result and Theorem 1.12 in [26] by Liu, Mubayi and Reiher, we can obtain the Turán density of the extension of $M_t^4$ ($t\ge 4$) and the degree stability (a stability stronger than the edge stability) of extremal hypergraphs. Combining our result and Theorem 1.4 in [24] by Keevash, Lenz and Mubayi [24], we can also obtain that if G is an $M_t^4$-free 4-uniform hypergraph with n vertices, then the α-spectral radius of G is no more than the α-spectral radius of $K_{4t-1}^4$ if $\alpha >1$, n is large enough and $t\ge 4$. Indeed, for hypergraphs satisfying the conditions in Theorem 1.12 by Liu, Mubayi and Reiher [26] or Theorem 1.4 in [24] by Keevash, Lenz and Mubayi, to obtain the degree stability of corresponding extremal hypergraphs or corresponding α-spectral results, it is sufficient to determine the Lagrangian densities of corresponding hypergraphs. These connections add more motivations to determine Lagrangian densities of hypergraphs.