{
  "id": "1303.2103",
  "version": "v4",
  "published": "2013-03-08T20:45:55.000Z",
  "updated": "2016-09-05T00:14:42.000Z",
  "title": "Exactly $m$-coloured complete infinite subgraphs",
  "authors": [
    "Bhargav Narayanan"
  ],
  "comment": "12 pages, fixed misprints, Journal of Combinatorial Theory, Series B",
  "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. The question of finding exactly $m$-coloured complete subgraphs was first considered by Erickson in 1994; in 1999, Stacey and Weidl partially settled a conjecture made by Erickson and raised some further questions. In this paper, we shall study, for a colouring of the edges of the complete graph on $\\mathbb{N}$ with exactly $k$ colours, how small the set of natural numbers $m$ for which there exists an $m$-coloured complete infinite subgraph can be. We prove that this set must have size at least $\\sqrt{2k}$; this bound is tight for infinitely many values of $k$. We also obtain a version of this result for colourings that use infinitely many colours.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-05-27T16:52:45.000Z",
      "title": "Exactly m-Coloured Complete Infinite Subgraphs",
      "abstract": "We say a graph is exactly m-coloured if we have a surjective map from the edges to some set of m colours. The question of finding exactly m-coloured complete subgraphs was first considered in 1994 by Erickson; in 1999, Stacey and Weidl partially settled a conjecture made by Erickson and raised some further questions. In this paper, we shall study, for a colouring of the edges of the complete graph on the natural numbers with exactly k colours, how small the set of natural numbers m for which there exists an exactly m-coloured complete infinite subgraph can be. We prove that this set must have size at least (2k)^(1/2); this bound is tight for infinitely many values of k. We also obtain a version of this result for colourings that use infinitely many colours.",
      "comment": "11 pages, submitted, added a new section on upper bound constructions",
      "journal": null,
      "doi": null,
      "authors": [
        "Bhargav P. Narayanan"
      ]
    },
    {
      "version": "v4",
      "updated": "2016-09-05T00:14:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C55",
      "05C63",
      "05D10"
    ],
    "keywords": [
      "exactly m-coloured complete infinite subgraph",
      "exactly m-coloured complete subgraphs",
      "natural numbers",
      "complete graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.2103N"
    }
  }
}