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Density of rainbow triangles and properly colored $K_4$'s

József Balogh, Peter Bradshaw, Ramon I. Garcia, Bernard Lidický

TL;DR

This work resolves sharp extremal questions for rainbow triangles and properly colored $K_4$’s in edge-colored graphs with fixed color-edge counts. It delivers a computer-free flag-algebra proof of the bound $T^2 \le 2RGB$, together with elementary counting and entropy proofs, and proves the uniqueness of the extremal balanced blowup of a properly colored $K_4$. It extends the methodology to show sharp bounds for the number of properly colored $K_4$’s, $K \le \tfrac{1}{4}\min\{RG,GB,RB\}$ and $K \le \tfrac{1}{4}(RGB)^{2/3}$, with equality characterizing balanced blowups. The paper also bounds the number of rainbow copies of $K_4$ under a fixed rainbow coloring, establishing $H \le \prod_{i=1}^6 C_i$, and provides three parallel proofs. Across sections, the authors tie flag-algebra technology to counting and entropy methods, emphasize blowup-translation between finite graphs and density limits, and discuss stability and potential extensions in incidence geometry contexts.

Abstract

T.-W. Chao and H.-H. H. Yu showed in 2023 that a graph with $R$ red, $G$ green, and $B$ blue edges has at most $\sqrt{2 RGB}$ rainbow triangles. They proved this bound using the entropy method. We give a computer-free flag-algebra proof of this bound, and we also convert our proof into a classical counting proof. The ideas in our proof lead to an even shorter entropy proof. We also show uniqueness of the extremal construction. Additionally, we prove a similar result that gives a sharp upper bound on the number of properly $3$-edge-colored $K_4$'s in graphs with $R$ red, $G$ green and $B$ blue edges.

Density of rainbow triangles and properly colored $K_4$'s

TL;DR

This work resolves sharp extremal questions for rainbow triangles and properly colored ’s in edge-colored graphs with fixed color-edge counts. It delivers a computer-free flag-algebra proof of the bound , together with elementary counting and entropy proofs, and proves the uniqueness of the extremal balanced blowup of a properly colored . It extends the methodology to show sharp bounds for the number of properly colored ’s, and , with equality characterizing balanced blowups. The paper also bounds the number of rainbow copies of under a fixed rainbow coloring, establishing , and provides three parallel proofs. Across sections, the authors tie flag-algebra technology to counting and entropy methods, emphasize blowup-translation between finite graphs and density limits, and discuss stability and potential extensions in incidence geometry contexts.

Abstract

T.-W. Chao and H.-H. H. Yu showed in 2023 that a graph with red, green, and blue edges has at most rainbow triangles. They proved this bound using the entropy method. We give a computer-free flag-algebra proof of this bound, and we also convert our proof into a classical counting proof. The ideas in our proof lead to an even shorter entropy proof. We also show uniqueness of the extremal construction. Additionally, we prove a similar result that gives a sharp upper bound on the number of properly -edge-colored 's in graphs with red, green and blue edges.

Paper Structure

This paper contains 6 sections, 9 theorems, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proofs of Theorem \ref{['triangleent']} via flag algebras, counting and entropy
  3. Proof of Theorem \ref{['thm:K4-counting']}
  4. Proof of Theorem \ref{['thm:fixedrainbowK4']}
  5. Concluding Remarks
  6. Acknowledgments

Key Result

Theorem 1.1

Let $\Gamma=(V,E)$ be a simple graph, and let the edges of $\Gamma$ be colored with red, green, and blue. Let $R$, $G$, $B$ denote the number of red, green, blue edges, respectively, and $T$ the number of rainbow triangles in $\Gamma$. Then, $T^2\leq 2RGB$.

Figures (4)

  • Figure 1: (a) A blowup of a properly $3$-edge-colored $K_4$. (b) An iterated blowup of a properly $3$-edge-colored $K_4$.
  • Figure 2: Two iterated constructions of a $6$-edge-colored complete graph where rainbow $K_4$ has density $24/215$.
  • Figure 3: Members of sets $S$ and $S'$ from Lemma \ref{['lem:gxb']}.
  • Figure 4: A drawing of $\Gamma[v_1,v_2,v_3,v_4]$ and the graph obtained by adding the resampled vertices $v_1',v_2'$.

Theorems & Definitions (26)

  • Theorem 1.1: Chao and Hans Yu ChaoEntropy2024
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['triangleent']}
  • Lemma 2.3
  • ...and 16 more