{
  "id": "2409.03222",
  "version": "v1",
  "published": "2024-09-05T03:37:08.000Z",
  "updated": "2024-09-05T03:37:08.000Z",
  "title": "On the size of sets avoiding a general structure",
  "authors": [
    "Runze Wang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a finite abelian group $G$ and a subset $S\\subseteq G$, we let $N_{G,\\ S}$ be the smallest integer $N$ such that for any subset $A\\subseteq G$ with $N$ elements, we have $g+S\\subseteq A$ for some $g\\in G$. Using the probabilistic method, we prove that \\begin{align*} \\frac{|H_G(S)|-1}{|H_G(S)|}|G|+\\Biggl\\lceil\\biggl(\\frac{|G|}{|H_G(S)|}\\biggr)^{1-|H_G(S)|/|S|}\\Biggr\\rceil\\le N_{G,\\ S}\\le \\biggl\\lfloor\\frac{|S|-1}{|S|}|G|\\biggr\\rfloor+1, \\end{align*} where $H_G(S)$ is the stabilizer of $S$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-05T03:37:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "general structure",
      "sets avoiding",
      "finite abelian group",
      "probabilistic method",
      "smallest integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}