We consider finite simple graphs. Given a graph H and a positive integer $n,$ the Turán number of H for the order $n,$ denoted $\mathrm {ex}(n,H),$ is the maximum size of a graph of order n not containing H as a subgraph. Erdős asked: ‘For which graphs H is it true that every graph on n vertices and $\mathrm {ex}(n,H)+1$ edges contains at least two H’s? Perhaps this is always true.’ We solve this problem in the negative by proving that for every integer $k\ge 4$ there exists a graph H of order k and at least two orders n such that there exists a graph of order n and size $\mathrm {ex}(n,H)+1$ which contains exactly one copy of $H.$ Denote by $C_4$ the $4$ -cycle. We also prove that for every integer n with $6\le n\le 11$ there exists a graph of order n and size $\mathrm {ex}(n,C_4)+1$ which contains exactly one copy of $C_4,$ but, for $n=12$ or $n=13,$ the minimum number of copies of $C_4$ in a graph of order n and size $\mathrm {ex}(n,C_4)+1$ is two.

我们考虑有限简单图。给定一个图H和一个正整数 $n ,$ 表示$n,$的阶H的 Turán 数 $\mathrm {ex}(n,H),$ 是不包含的n阶图的最大尺寸H作为子图。Erdős 问道：“对于哪些图H ， n个顶点和 $\mathrm {ex}(n,H)+1$ 边上的每个图都包含至少两个H是真的吗？也许这总是正确的。我们通过证明对于每个整数 $k\ge 4$ 都存在一个阶图H来否定地解决这个问题 k和至少两个阶n使得存在阶n和大小 $\mathrm {ex}(n,H)+1$ 的图，其中恰好包含 $H 的一个副本。$用 $C_4$ 表示 $4$ -循环。我们还证明，对于每个具有 $6\le n\le 11$ 的整数n ，存在一个阶n且大小 为 $\mathrm {ex}(n,C_4)+1$ 的图，其中恰好包含 $C_4,$ 的一个副本但是，对于 $n=12$ 或 $n=13,$ n 阶图中 $C_4$ 的最小副本数 并且大小 $\mathrm {ex}(n,C_4)+1$ 是二。