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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 ofErd\H{o}s gives a strengthening: there are not only one, but at least$\lfloor\frac{n}{2}\rfloor$ triangles. We give a further improvement: if thereis no vertex contained by all triangles then there are at least $n-2$ of them.There are some natural generalizations when $(a)$ complete graphs areconsidered (rather than triangles), $(b)$ the graph has $t$ extra edges (notonly one) or $(c)$ it is supposed that there are no $s$ vertices such thatevery triangle contains one of them. We were not able to prove thesegeneralizations, they are posed as conjectures.

The number of triangles is more when they have no common vertex