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Maximum number of edge colorings avoiding rainbow copies of $K_4$

Hiêp Hàn, Carlos Hoppen, Nicolas Moro Müller, Dionatan Ricardo Schmidt

TL;DR

The paper resolves the number of rainbow-$K_4$-free $r$-edge-colorings on large graphs, showing that for every $r\ge 12$ and sufficiently large $n$, the maximum is $r^{\mathrm{ex}(n,K_4)}$, achieved only by the Turán graph $T_3(n)$. The authors combine a stability argument with a hypergraph container framework based on multicolor templates to constrain the structure of extremal colorings, first showing near-tripartite structure and then invoking an exactness result to obtain the Turán graph as the unique extremal graph. This advances the generalized Erdős-Rothschild program for rainbow patterns by providing tight bounds at a sharply reduced color threshold and confirming a conjecture of Gupta, Pehova, Powierski, and Staden. The methods establish a versatile template-container approach for counting pattern-avoiding colorings in graphs, with potential applicability to other rainbow-avoidance problems. The result tightens previously known bounds and contributes to the broader understanding of how multipartite structure governs extremal rainbow-coloring phenomena.

Abstract

In this paper we show that for $r\geq 12$ and any sufficiently large $n$-vertex graph $G$ the number of $r$-edge-colorings of $G$ with no rainbow $K_4$ is at most $r^{ex(n,K_4)}$, where $ex(n,K_4)$ denotes the Turán number of $K_4$. Moreover, $G$ attains equality if and only if it is the Turán graph $T_3(n)$. The bound on the number of colors $r\geq 12$ is best possible. It improves upon a result of H. Lefmann, D.A. Nolibos, and the second author who showed the same result for $r \geq 5434$ and it confirms a conjecture by Gupta, Pehova, Powierski and Staden.

Maximum number of edge colorings avoiding rainbow copies of $K_4$

TL;DR

The paper resolves the number of rainbow--free -edge-colorings on large graphs, showing that for every and sufficiently large , the maximum is , achieved only by the Turán graph . The authors combine a stability argument with a hypergraph container framework based on multicolor templates to constrain the structure of extremal colorings, first showing near-tripartite structure and then invoking an exactness result to obtain the Turán graph as the unique extremal graph. This advances the generalized Erdős-Rothschild program for rainbow patterns by providing tight bounds at a sharply reduced color threshold and confirming a conjecture of Gupta, Pehova, Powierski, and Staden. The methods establish a versatile template-container approach for counting pattern-avoiding colorings in graphs, with potential applicability to other rainbow-avoidance problems. The result tightens previously known bounds and contributes to the broader understanding of how multipartite structure governs extremal rainbow-coloring phenomena.

Abstract

In this paper we show that for and any sufficiently large -vertex graph the number of -edge-colorings of with no rainbow is at most , where denotes the Turán number of . Moreover, attains equality if and only if it is the Turán graph . The bound on the number of colors is best possible. It improves upon a result of H. Lefmann, D.A. Nolibos, and the second author who showed the same result for and it confirms a conjecture by Gupta, Pehova, Powierski and Staden.

Paper Structure

This paper contains 5 sections, 8 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Notation and Tools
  3. The hypergraph container theorem
  4. Multicolor containers
  5. Proof of Theorem \ref{['thm_main2']}

Key Result

Theorem 1.1

rainbow_kn For every $k \geq 3$ and $r \geq \binom{k}{2}^{8k-4}$ there is $n_0$ such that every graph of order $n > n_0$ has at most $r^{\mathop{\mathrm{ex}}\nolimits(n,K_{k})}$ rainbow-$K_{k}$-free $r$-colorings. Moreover, the Turán graph $T_{k-1}(n)$ is the only graph on $n$ vertices which achieve

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2: BBCLMS
  • Theorem 2.3
  • Definition 2.4: Template, subtemplate and rainbow copies
  • Theorem 2.5
  • proof
  • ...and 14 more