In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal–Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a 3-edge-colored graph with red, green, blue edges, the number of rainbow triangles is at most , which is sharp. Second, we give a generalization of the Kruskal–Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.

中文翻译：

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通过熵方法求解 Kruskal-Katona 型问题


在本文中，我们研究了几个极值组合问题，这些问题要求给定边数的固定子图的最大副本数。我们将这种类型的问题称为克鲁斯卡尔-卡托纳型问题。本文将讨论的大多数问题都与关节问题有关。本文有两个主要结果。首先，我们证明，在具有红、绿、蓝边的三边彩色图中，彩虹三角形的数量最多为 ，这是尖锐的。其次，我们给出了克鲁斯卡尔-卡托纳定理的概括，这意味着许多其他先前的概括。这两个论证都使用了熵方法，主要创新在于一个更聪明的论证，它改善了希勒不等式给出的界限。