{
  "id": "1710.08352",
  "version": "v1",
  "published": "2017-10-23T15:53:03.000Z",
  "updated": "2017-10-23T15:53:03.000Z",
  "title": "Maximum number of sum-free colorings in finite abelian groups",
  "authors": [
    "Hiep Hàn",
    "Andrea Jiménez"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "An $r$-coloring of a subset $A$ of a finite abelian group $G$ is called sum-free if it does not induce a monochromatic Schur triple, i.e., a triple of elements $a,b,c\\in A$ with $a+b=c$. We investigate $\\kappa_{r,G}$, the maximum number of sum-free $r$-colorings admitted by subsets of $G$, and our results show a close relationship between $\\kappa_{r,G}$ and largest sum-free sets of $G$. Given a sufficiently large abelian group $G$ of type $I$, i.e., $|G|$ has a prime divisor $q$ with $q\\equiv 2\\pmod 3$. For $r=2,3$ we show that a subset $A\\subset G$ achieves $\\kappa_{r,G}$ if and only if $A$ is a largest sum-free set of $G$. For even order $G$ the result extends to $r=4,5$, where the phenomenon persists only if $G$ has a unique largest sum-free set. On the contrary, if the largest sum-free set in $G$ is not unique then $A$ attains $\\kappa_{r,G}$ if and only if it is the union of two largest sum-free sets (in case $r=4$) and the union of three (\"independent\") largest sum-free sets (in case $r=5$). Our approach relies on the so called container method and can be extended to larger $r$ in case $G$ is of even order and contains sufficiently many largest sum-free sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-23T15:53:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "05C35",
      "05D99",
      "05E15"
    ],
    "keywords": [
      "finite abelian group",
      "maximum number",
      "sum-free colorings",
      "unique largest sum-free set",
      "monochromatic schur triple"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}