Toward a rainbow Corrádi--Hajnal Theorem \RNum{1}

TL;DR

This work advances the anti-Ramsey theory for rainbow triangle packings by resolving the first interval of the conjectured landscape: for large $n$ and $t o [1,(2n-6)/9 - 2]$, the minimum number of colors needed to force a rainbow $tK_3$ in any edge-coloring of $K_n$ equals $ ext{Xi}(n,t)+2$, with $ ext{Xi}(n,t)$ matching the edge- counts of five extremal constructions. The authors develop a stability framework around a maximal tiling of $(t+1)K_3$-free graphs, and prove that any near-extremal representative must contain a large rainbow tripartite subgraph. By combining stability with a large rainbow tripartite subgraph, they derive a contradiction to the absence of a rainbow $(t+2)K_3$, thereby proving the conjectured formula for the first interval. The results improve prior bounds and provide a pathway to extend the analysis to the remaining intervals, where the asymptotic behavior of $ ext{ar}(n,(t+2)K_3)$ is tied to known Turán-type extremal numbers. Overall, the paper blends extremal graph theory with anti-Ramsey coloring methods to sharpen the rainbow packing thresholds in $K_n$.

Abstract

We study an anti-Ramsey extension of the classical Corrádi--Hajnal Theorem: how many colors are needed to color the complete graph on $n$ vertices in order to guarantee a rainbow copy of $t K_{3}$, that is, $t$ vertex-disjoint triangles. We provide a conjecture for large $n$, consisting of five classes of different extremal constructions, corresponding to five subintervals of $\left[1,\, \tfrac{n}{3}\right]$ for the parameter $t$. In this work, we establish this conjecture for the first interval, $t \in \left[1,\, \tfrac{2n-6}{9}\right]$. In particular, this improves upon a recent result of Lu--Luo--Ma~[arXiv:2506.07115] which established the case $t \le \tfrac{n - 57}{15}$.

Figures (8)

• Figure 1: Constructions $E_1(n,t),\dots,E_5(n,t)$.
• Figure 2: The extremal graphs.
• Figure 3: Members in $\mathcal{T}_1$ and $\mathcal{T}_2$.
• Figure 4: $z_1y_2,z_1z_2 \in |H |$.
• Figure 5: $|\{\omega_1,\omega_2,\omega_3\}\cap V(\mathcal{M})|=0$.

Theorems & Definitions (37)

• Conjecture 1.2
• Theorem 1.3: DHLY25
• Theorem 1.4
• Theorem 1.5: ABHP15
• Lemma 2.1: ABHP15
• Lemma 2.4: ABHP15
• Lemma 2.5: ABHP15
• Theorem 2.6: AES74
• Theorem 3.1
• proof : Proof of Theorem \ref{['THM:STABILITY']}