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

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 Lovász (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 $n\ge kr+\lfloor \frac{k-1}{r+1}\rfloor$. 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.

5 pages. Summary paper of a problem solved at the Emlektabla Conference 2010. Reference added and small note made about the previous conjectures