By the theorem of Mantel (1907) 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 $\left(a\right)$ complete graphs are considered (rather than triangles), $\left(b\right)$ the graph has $t$ extra edges (not only one) or $\left(c\right)$ 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.

根据Mantel（1907）的定理，已知 $ñ$ 顶点和 $\lfloor \frac{{ñ}^2}{4}\rfloor +\mathrm{1个}$边必须包含三角形。埃尔德斯定理给出了一种加强：不仅存在一个定理，而且至少存在一个定理$\lfloor \frac{ñ}{2}\rfloor$三角形。我们给出了进一步的改进：如果所有三角形都没有包含顶点，那么至少存在$ñ-2$其中。有一些自然的概括$（\mathrm{一个}）$ 考虑完整的图形（而不是三角形）， $（b）$ 该图有 $Ť$ 额外的边缘（不仅是一个）或 $（C）$ 假设没有 $s$顶点，使得每个三角形都包含其中一个。我们无法证明这些概括，它们被当作推测。