{
  "id": "1902.09916",
  "version": "v1",
  "published": "2019-02-26T13:16:39.000Z",
  "updated": "2019-02-26T13:16:39.000Z",
  "title": "Finite sums of arithmetic progressions",
  "authors": [
    "Shahram Mohsenipour"
  ],
  "comment": "Submitted",
  "categories": [
    "math.CO"
  ],
  "abstract": "For two $l$-term arithmetic progressions $P$ and $Q$, we define their pointwise sum $P\\oplus Q$ as the $l$-term arithmetic progression whose $i$th term is the sum of the $i$th terms of $P$ and $Q$. So we can talk about finite sums of $l$-term arithmetic progressions. In this paper we give a combination of van der Waerden's theorem and Hindman's theorem by showing that for positive integers $c$ and $l\\geq3$, if $\\mathbb{N}$ is $c$-colored, then there are infinitely many $l$-term arithmetic progressions $P_i$, $i\\in\\mathbb{N}$ such that all of their finite sums (with no repetition) are monochromatic with the same color, call it $\\gamma$. We can do more by showing that all the common differences of the finite sums of $P_i$ can have also the color $\\gamma$, thus a combination of the van der Waerden-Brauer theorem and Hindman's theorem. We also give a tower bound for the finite case of the first mentioned theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-26T13:16:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite sums",
      "term arithmetic progression",
      "hindmans theorem",
      "van der waerdens theorem",
      "th term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}