在最大度条件下广义三角形的正共度极值问题研究

Positive Codegree Andrásfai-Erdős-Sós Theorem for the Generalized Triangle under Max-Degree Constraints

摘要: 著名的Andrásfai-Erdős-Sós定理表明：任意一个n个顶点、最小度大于2n/5的三角形禁止图必为二部图。设$T_r=\left\{\left\{1,\cdots ,r-1,r\right\},\left\{1,\cdots ,r-1,r+1\right\},\left\{r,r+1,\cdots ,2r-1\right\}\right\}$ 为r-一致的广义三角形。在本文中，我们证明了：当$n$ 充分大时，若$\mathscr{H}$ 是一个含n个顶点的$T_r$ -禁止的r-一致超图，并且满足每一个$\left(r-1\right)$ -元组要么不包含于任何超边中，要么至少包含于超过$\frac{n}{r}-\frac{{\Delta }_{r-1}\left(\mathscr{H}\right)}{2r}$ 条超边中，则$\mathscr{H}$ 必为r-部超图。

Abstract: The celebrated Andrásfai-Erdős-Sós Theorem shows that every n-vertex triangle-free graph with minimum degree greater than 2n/5 must be bipartite. Denote the $T_r=\left\{\left\{1,\cdots ,r-1,r\right\},\left\{1,\cdots ,r-1,r+1\right\},\left\{r,r+1,\cdots ,2r-1\right\}\right\}$ r-uniform generalized triangle. In this paper, we prove the following strengthening of the above theorem: for sufficiently large n, if $\mathscr{H}$ is an n-vertex $T_r$ -free r-uniform hypergraph such that every $\left(r-1\right)$ -tuple of vertices is either contained in no hyperedge or in more than $\frac{n}{r}-\frac{{\Delta }_{r-1}\left(\mathscr{H}\right)}{2r}$ hyperedges, then $\mathscr{H}$ must be r-partite.