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 this type of problems 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 $3$-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学问题，这类问题要求给定边数条件下固定子图的最大拷贝数，我们称之为Kruskal--Katona型问题。文中讨论的大部分问题与关节问题相关。本文包含两个主要结果：首先，我们证明在具有R条红边、G条绿边、B条蓝边的三色图中，彩虹三角形的数量至多为√(2RGB)，且该上界是紧的；其次，我们给出Kruskal--Katona定理的一个推广，该推广蕴含了此前多个相关推广结果。两个论证均使用熵方法，其核心创新在于通过更巧妙的论证改进了Shearer不等式给出的上界。