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

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 $[1,\frac{n}{3}]$ for the parameter $t$. In this work, we establish this conjecture for the first interval, $t\in [1,\frac{2n-6}{9}]$. In particular, this improves upon a recent result of Lu–Luo–Ma~[arXiv:2506.07115] which established the case $t\le \frac{n-57}{15}$.