{
  "id": "1310.1386",
  "version": "v2",
  "published": "2013-10-04T19:59:15.000Z",
  "updated": "2016-09-06T13:27:06.000Z",
  "title": "Approximations to $m$-coloured complete infinite hypergraphs",
  "authors": [
    "Teeradej Kittipassorn",
    "Bhargav Narayanan"
  ],
  "comment": "12 pages, fixed misprints, Journal of Graph Theory",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. In 1999, Stacey and Weidl, partially resolving a conjecture of Erickson from 1994, showed that for a fixed natural number $m>2$ and for all sufficiently large $k$, there is a $k$-colouring of the complete graph on $\\mathbb{N}$ such that no complete infinite subgraph is exactly $m$-coloured. In the light of this result, we consider the question of how close we can come to finding an exactly $m$-coloured complete infinite subgraph. We show that for a natural number $m$ and any finite colouring of the edges of the complete graph on $\\mathbb{N}$ with $m$ or more colours, there is an exactly ${\\hat m}$-coloured complete infinite subgraph for some ${\\hat m}$ satisfying $|m-{\\hat m}|\\le \\sqrt{m/2} + 1/2$; this is best-possible up to the additive constant. We also obtain analogous results for this problem in the setting of $r$-uniform hypergraphs. Along the way, we also prove a recent conjecture of the second author and investigate generalisations of this conjecture to $r$-uniform hypergraphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-04T19:59:15.000Z",
      "title": "Approximations to m-Coloured Complete Infinite Subgraphs",
      "abstract": "Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. In 1996, Stacey and Weidl, partially resolving a conjecture of Erickson from 1994, showed that for a fixed natural number m>2 and for all sufficiently large k, there is a k-colouring of complete graph on N such that no complete infinite subgraph is exactly m-coloured. In the light of this result, we consider the question of how close we can come to finding an exactly m-coloured complete infinite subgraph. We show that for a natural number m and any finite colouring of the edges of the complete graph on N with m or more colours, there is an exactly m'-coloured complete infinite subgraph for some m' satisfying |m-m'| <= (m/2)^(1/2) + 1/2; this is best possible up to an additive constant. We also obtain analogous results for this problem in the setting of r-uniform hypergraphs. Along the way, we also prove a recent conjecture of the second author and investigate generalisations of this conjecture to r-uniform hypergraphs.",
      "comment": "9 pages, submitted",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-09-06T13:27:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C55",
      "05C63",
      "05C65",
      "05D10"
    ],
    "keywords": [
      "r-uniform hypergraphs",
      "approximations",
      "complete graph",
      "conjecture",
      "exactly m-coloured complete infinite subgraph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.1386K"
    }
  }
}