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 $R$ red, $G$ green, $B$ blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$, 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.

翻译：本文研究了若干极值组合问题，这些问题的核心是在给定边数的条件下，固定子图的最大可能数量。我们将此类问题称为Kruskal-Katona型问题。本文讨论的大多数问题都与关节问题相关。本文包含两项主要结果：首先，我们证明了在具有$R$条红边、$G$条绿边、$B$条蓝边的三边着色图中，彩虹三角形数量的上界为$\sqrt{2RGB}$，且该上界是紧的；其次，我们提出了Kruskal-Katona定理的一个推广形式，该形式蕴含了许多先前已有的推广。两项证明均采用熵方法，其主要创新点在于通过更精巧的论证改进了由Shearer不等式给出的界。