论文标题
当三角形没有共同的顶点时，三角形的数量会更多
The number of triangles is more when they have no common vertex
论文作者
论文摘要
通过Mantel $ [5] $的定理，众所周知，具有$ n $顶点的图和$ \ lfloor \ frac {n^{2}}} {4} {4} \ rfloor+1 $边缘必须包含一个三角形。 Erdős的定理给出了一个增强：不仅有一个，而且至少有$ \ lfloor \ frac {n} {2} {2} \ rfloor $ triangles。我们给予进一步的进步：如果没有所有三角形都包含的顶点，那么其中至少有$ n-2 $。当考虑$（a）$完整的图（而不是三角形）时，有一些自然概括，$（b）$该图具有$ t $额外的边缘（不仅是一个）或$（c）$，因此假设没有$ s $ pertices，这样每个三角形都包含其中一个。我们无法证明这些概括，它们被认为是猜想。
By the theorem of Mantel $[5]$ it is known that a graph with $n$ vertices and $\lfloor \frac{n^{2}}{4} \rfloor+1$ edges must contain a triangle. A theorem of Erdős gives a strengthening: there are not only one, but at least $\lfloor\frac{n}{2}\rfloor$ triangles. We give a further improvement: if there is no vertex contained by all triangles then there are at least $n-2$ of them. There are some natural generalizations when $(a)$ complete graphs are considered (rather than triangles), $(b)$ the graph has $t$ extra edges (not only one) or $(c)$ it is supposed that there are no $s$ vertices such that every triangle contains one of them. We were not able to prove these generalizations, they are posed as conjectures.
