Generalized Turán problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing the number of cliques of a fixed order in a graph with fixed number of vertices and bounded maximum degree, was recently completely resolved by Chase. Kirsch and Radcliffe raised a natural variant of this problem where the number of edges is fixed instead of the number of vertices. In this paper, we determine the maximum number of cliques of a fixed order in a graph with fixed number of edges and bounded maximum degree, resolving a conjecture by Kirsch and Radcliffe. We also give a complete characterization of the extremal graphs.

在过去的几十年中，广义图兰问题一直是极值组合学研究的中心课题。最近，Chase 完全解决了这样一个问题，即最大化具有固定顶点数和有界最大度的图中的固定顺序团的数量。Kirsch 和 Radcliffe 提出了这个问题的一个自然变体，其中边的数量是固定的，而不是顶点的数量。在本文中，我们确定具有固定边数和有界最大度的图中固定阶数的最大团数，解决了 Kirsch 和 Radcliffe 的猜想。我们还给出了极值图的完整表征。