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

TL;DR

This paper proves that in any 3-coloring of the complete 3-uniform hypergraph on 12 vertices, there always exists a 2-colored matching of size 4, advancing understanding of monochromatic matchings in hypergraph colorings.

Contribution

It confirms a specific case of a conjecture about 2-colored matchings in 3-colored complete hypergraphs, providing a concrete proof for the case of n=12, r=3, k=4.

Findings

In every 3-coloring of K_{12}^3, a 2-colored matching of size 4 always exists.

Supports the conjecture relating hypergraph size, coloring, and matching size.

Advances the understanding of monochromatic structures in hypergraph Ramsey theory.

Abstract

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 $n \geq kr + \lfloor \frac{k-1}{r+1} \rfloor$. The smallest test case is when $r=3$ and $k=4$. We prove that in every 3-coloring of the edges of $K_{12}^3$ there is a 2-colored matching of size 4.