图的Sylvester–Gallai维数
The Sylvester--Gallai dimension of graphs
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中文总结 AI 辅助
本文研究新图参数Sylvester–Gallai维数，证明了有界度图、稀疏图等图族的该参数下界，及随机图的该参数上界。
中文摘要 AI 辅助
对于无向图$G$，其\textit{Sylvester–Gallai维数}记为$\text{SGdim}(G)$，是由$V(G)$索引的不同实点构型的最大仿射维数，其中由$G$的边确定的每条线均为特殊线（包含至少三个点）。因此经典的Sylvester–Gallai定理可表述为完全图$K_n$满足$\text{SGdim}(K_n)=1$。本文对这一新图参数展开系统研究，证明了特定图族（有界度图、稀疏图、子图无关图）的下界，以及随机图的上界。
英文摘要
For an undirected graph $G$, its \emph{Sylvester--Gallai dimension} $\text{SGdim}(G)$ is the largest affine dimension of a configuration of distinct real points indexed by $V(G)$ in which every line determined by an edge of $G$ is a special line (contains at least three points). Hence, the classical Sylvester--Gallai theorem can be stated as $\text{SGdim}(K_n)=1$ for the complete graph $K_n$. We initiate the systematic study of this new graph parameter and prove lower bounds for certain graph families (bounded degree, sparse, minor-free) as well as an upper bound for random graphs.
