2-Colored Matchings in a 3-Colored K^{3}_{12}

Let $K_n^r$ denote the complete $r$-uniform hypergraph on $n$ vertices. A matching $M$ in a hypergraph is a set of pairwise vertex disjoint edges. Recent Ramsey-type results rely on lemmas about the size of monochromatic matchings. A starting point for this study comes from a well-known result of Alon, Frankl, and Lov'asz (1986). Our motivation is to find the smallest $n$ such that every $t$-coloring of $K_n^r$ contains an $s$-colored matching of size $k$. It has been conjectured that in every coloring of the edges of $K_n^r$ with 3 colors there is a 2-colored matching of size at least $k$ provided that $n \geq kr + \lfloor \frac{k-1}{r+1} \rfloor$\mathrm{.}Thesmallesttestcaseiswhen$r=3$and$k=4$\mathrm{.}Weprovethatinevery3-coloringoftheedgesof$K_{12}^3$ there is a 2-colored matching of size 4.