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：本文研究了几类极值组合学问题，这些问题探讨在给定边数条件下固定子图的最大拷贝数。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联合问题相关。本文主要有两个结果：首先，我们证明在一个包含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，我们给出了Kruskal--Katona定理的一个推广形式，该推广蕴含了先前诸多其他推广结论。两个论证均采用熵方法，主要创新点在于通过更巧妙的论证改进了由Shearer不等式给出的上界。