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A Purely Entropic Approach to the Rainbow Triangle Problem

Ting-Wei Chao, Hung-Hsun Hans Yu

TL;DR

The paper proves a tight bound on the number of rainbow triangles in a 3-edge-colored simple graph by a purely entropic method. It stays within the entropy framework throughout, introducing a sampling scheme and a novel key injection that compresses the relevant information. The injection enables a clean entropy bound that leads to $T^2 \le 2RGB$, equivalently $T \le \sqrt{2RGB}$. This approach clarifies entropy-based reasoning for rainbow-triangle problems and suggests connections to flag-algebra ideas and possible generalizations in related combinatorial problems.

Abstract

In this short note, we present a purely entropic proof that in a $3$-edge-colored simple graph with $R$ red edges, $G$ green edges, and $B$ blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$.

A Purely Entropic Approach to the Rainbow Triangle Problem

TL;DR

The paper proves a tight bound on the number of rainbow triangles in a 3-edge-colored simple graph by a purely entropic method. It stays within the entropy framework throughout, introducing a sampling scheme and a novel key injection that compresses the relevant information. The injection enables a clean entropy bound that leads to , equivalently . This approach clarifies entropy-based reasoning for rainbow-triangle problems and suggests connections to flag-algebra ideas and possible generalizations in related combinatorial problems.

Abstract

In this short note, we present a purely entropic proof that in a -edge-colored simple graph with red edges, green edges, and blue edges, the number of rainbow triangles is at most .
Paper Structure (6 sections, 2 theorems, 19 equations, 5 figures)

This paper contains 6 sections, 2 theorems, 19 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main Proof
  3. Sampling random variables
  4. Key injection
  5. Proof of the key injection
  6. A concluding remark

Key Result

Theorem 1.1

Let $G=(V,E)$ be a simple graph, and each edge is colored in one of the three colors: red, green, or blue. Suppose there are $R$ red edges, $G$ green edges, $B$ blue edges, and $T$ rainbow triangles. Then we have

Figures (5)

  • Figure 1: $v'_r$ is either $v_r$ or $u$.
  • Figure 2: The definition of $T_x$.
  • Figure 3: The case when $z=y$ and $(x,z_1,z_2)$ forms a rainbow triangle in order.
  • Figure 4: The case when $z=y$ and $(x,z_2,z_1)$ forms a rainbow triangle in order.
  • Figure 5: The remaining case.

Theorems & Definitions (4)

  • Theorem 1.1: CY23
  • Lemma 2.1
  • proof : Proof of \ref{['thm:RT']} assuming \ref{['lemma:injection']}
  • proof : Proof of \ref{['lemma:injection']}