arXiv 1209.20332012-09-13math.CO

2-三色K^{3}_{12}中的色匹配

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

Neal Bushaw, Peter Csorba, Lindsay Erickson, Daniel Gerbner, Diana Piguet, Ago Riet, Tamas Terpai, Dominik Vu

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

摘要：

设$K_{n}^{r}$表示顶点为$n$的完全$r$ -一致超图.超图中的匹配$M$是成对顶点不相交的边的集合。最近的拉姆齐型结果依赖于关于单色匹配的大小的引理。本研究的出发点来自Alon、Frankl和Lovász（1986）的一个著名结果。我们的动机是找到最小的$n$，使得$K_{n}^{r}$的每个$t$ -着色都包含大小为$k$的$s$ -着色匹配.已经证实，在$K_n^r$的边缘的每一个着色中，如果$n \geq kr + \lfloor \frac{k-1}{r+1} \rfloor$具有3种颜色，则存在大小至少为$k$的2种颜色匹配。最小测试用例为$r=3$和$k=4$。我们证明了在$K_{12}^3$的边的每一个3-着色中，存在一个尺寸为4的2-色匹配。

英文摘要：

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 $n \geq kr + \lfloor \frac{k-1}{r+1} \rfloor$. 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.

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英文摘要：

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