Given a graph H and an integer p, the edge blow-up $H^{p+1}$ of H is the graph obtained from replacing each edge in H by a clique of order $p+1$ where the new vertices of the cliques are all distinct. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. In this paper, we determine the range of the Turán numbers for edge blow-up of all bipartite graphs and the exact Turán numbers for edge blow-up of all non-bipartite graphs. In addition, we characterize the extremal graphs for edge blow-up of all non-bipartite graphs. Our results also extend several known results, including the Turán numbers for edge blow-up of stars, paths and cycles. The method we used can also be applied to find a family of counter-examples to a conjecture posed by Keevash and Sudakov in 2004 concerning the maximum number of edges not contained in any monochromatic copy of H in a 2-edge-coloring of $K_n$.

给定一个图H和一个整数p，边缘爆炸$H^{磷+1}$的ħ是从替换每个边缘所获得的曲线图ħ通过的顺序的集团$磷+1$其中集团的新顶点都是不同的。Erdős和Moon首先研究了匹配边缘爆炸的图兰数。在本文中，我们确定了所有二部图边缘膨胀的图兰数的范围和所有非二部图边缘膨胀的确切图兰数。此外，我们表征了所有非二部图的边缘膨胀的极值图。我们的结果还扩展了几个已知结果，包括恒星边缘爆炸、路径和循环的图兰数。我们使用的方法也可以用于找到一系列反例，以解决 Keevash 和 Sudakov 在 2004 年提出的关于不包含在H 的任何单色副本中的最大边数的猜想的 2 边着色${钾}_n$.