{
  "id": "1209.2033",
  "version": "v2",
  "published": "2012-09-10T15:35:06.000Z",
  "updated": "2012-09-13T14:35:01.000Z",
  "title": "2-Colored Matchings in a 3-Colored K^{3}_{12}",
  "authors": [
    "Neal Bushaw",
    "Peter Csorba",
    "Lindsay Erickson",
    "Daniel Gerbner",
    "Diana Piguet",
    "Ago Riet",
    "Tamas Terpai",
    "Dominik Vu"
  ],
  "comment": "5 pages. Summary paper of a problem solved at the Emlektabla Conference 2010. Reference added and small note made about the previous conjectures",
  "categories": [
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-09-13T14:35:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pairwise vertex disjoint edges",
      "smallest test case",
      "ramsey-type results",
      "uniform hypergraph",
      "monochromatic matchings"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.2033B"
    }
  }
}