{
  "id": "math/0506351",
  "version": "v1",
  "published": "2005-06-17T13:55:15.000Z",
  "updated": "2005-06-17T13:55:15.000Z",
  "title": "On Monochromatic Ascending Waves",
  "authors": [
    "Tim LeSaulnier",
    "Aaron Robertson"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A sequence of positive integers $w_1,w_2,...,w_n$ is called an ascending wave if $w_{i+1}-w_i \\geq w_i - w_{i-1}$ for $2 \\leq i \\leq n-1$. For integers $k,r\\geq1$, let $AW(k;r)$ be the least positive integer such that under any $r$-coloring of $[1,AW(k;r)]$ there exists a $k$-term monochromatic ascending wave. The existence of $AW(k;r)$ is guaranteed by van der Waerden's theorem on arithmetic progressions since an arithmetic progression is, itself, an ascending wave. Originally, Brown, Erd\\H{o}s, and Freedman defined such sequences and proved that $k^2-k+1\\leq AW(k;2) \\leq {1/3}(k^3-4k+9)$. Alon and Spencer then showed that $AW(k;2) = O(k^3)$. In this article, we show that $AW(k;3) = O(k^5)$ as well as offer a proof of the existence of $AW(k;r)$ independent of van der Waerden's theorem. Furthermore, we prove that for any $\\epsilon > 0$, $$ \\frac{k^{2r-1-\\epsilon}}{2^{r-1}(40r)^{r^2-1}}(1+o(1)) \\leq AW(k;r) \\leq \\frac{k^{2r-1}}{(2r-1)!}(1+o(1)) $$ holds for all $r \\geq 1$, which, in particular, improves upon the best known upper bound for $AW(k;2)$. Additionally, we show that for fixed $k \\geq 3$, $$ AW(k;r)\\leq\\frac{2^{k-2}}{(k-1)!} r^{k-1}(1+o(1)). $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-06-17T13:55:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "van der waerdens theorem",
      "arithmetic progression",
      "term monochromatic ascending wave",
      "positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......6351L"
    }
  }
}