{
  "id": "math/0302155",
  "version": "v3",
  "published": "2003-02-13T04:22:10.000Z",
  "updated": "2003-02-22T19:05:31.000Z",
  "title": "Generalized additive bases, Konig's lemma, and the Erdos-Turan conjecture",
  "authors": [
    "Melvyn B. Nathanson"
  ],
  "comment": "8 pages. LaTex. Some new results have been added and some typos corrected",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let A be a set of nonnegative integers. For every nonnegative integer n and positive integer h, let r_{A}(n,h) denote the number of representations of n in the form n = a_1 + a_2 + ... + a_h, where a_1, a_2,..., a_h are elements of A and a_1 \\leq a_2 \\leq ... \\leq a_h. The infinite set A is called a basis of order h if r_{A}(n,h) \\geq 1 for every nonnegative integer n. Erdos and Turan conjectured that limsup_{n\\to\\infty} r_A(n,2) = \\infty for every basis A of order 2. This paper introduces a new class of additive bases and a general additive problem, a special case of which is the Erdos-Turan conjecture. Konig's lemma on the existence of infinite paths in certain graphs is used to prove that this general problem is equivalent to a related problem about finite sets of nonnegative integers.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2003-02-22T19:05:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B34",
      "11B05"
    ],
    "keywords": [
      "konigs lemma",
      "generalized additive bases",
      "erdos-turan conjecture",
      "nonnegative integer",
      "infinite set"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......2155N"
    }
  }
}