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Kruskal--Katona-Type Problems via the Entropy Method

Ting-Wei Chao, Hung-Hsun Hans Yu

Abstract

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.

Kruskal--Katona-Type Problems via the Entropy Method

Abstract

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 -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.
Paper Structure (7 sections, 13 theorems, 43 equations, 2 figures)

This paper contains 7 sections, 13 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rainbow triangle
  4. Proof of \ref{['theorem:RainbowTriangle']}
  5. Higher-uniformity analogue
  6. A generalization of Kruskal--Katona
  7. Entropy-based Proof for Kruskal--Katona

Key Result

Theorem 1.1

Let $G=(V,E)$ be a simple graph and let edges be colored in colors red, green, and blue, so that each edge receives exactly one color. Let $R,G,B$ denote the number of red, green, blue edges, respectively. Let $T$ be the number of rainbow triangles in the graph. Then $T^2\leq 2RGB$.

Figures (2)

  • Figure 1: $2N^2$ edges of each color with $4N^3$ rainbow triangles.
  • Figure 2: The random variables sampled in the proof of \ref{['theorem:RainbowTriangle']}.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2: Lovász's version of the Kruskal--Katona theorem
  • Theorem 1.3: Chowdhury--Patkós CP10
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3: Subadditivity
  • Theorem 2.4: Shearer's inequality
  • Theorem 2.5: A special case of Shearer's inequality
  • Definition 2.6
  • Proposition 2.7: Chain rule
  • ...and 13 more