{
  "abstract": "Fix integers $k,b,q$ with $k\\ge 2,b\\ge 0,q\\ge 2$. Define the function $p$ to be: $p(x)=kx+b$. We call a set $S$ of integers $(k,b,q)$-linear-free if $x\\in S$ implies $p^i (x)\\notin S$ for all $i=1,2,\\dots,q-1$, where $p^i (x)=p\\left(p^{i-1}(x)\\right)$ and $p^0 (x)=x$. Such a set $S$ is maximal in $[n]:=\\{1,2,\\dots,n\\}$ if $S\\cup \\{t\\},\\forall t\\in [n]\\backslash S$ is not $(k,b,q)$-linear-free. Let $M_{k,b,q} (n)$ be the set of all maximal $(k,b,q)$-linear-free subsets of $[n]$, and define $g_{k,b,q} (n)=\\min_{S\\in M_{k,b,q} (n)}|S|$ and $f_{k,b,q} (n)=\\max_{S\\in M_{k,b,q} (n)}|S|$. In this paper, formulae for $g_{k,b,q} (n)$ and $f_{k,b,q} (n)$ are proposed. Also, it is proven that there is at least one maximal $(k,b,q)$-linear-free subset of $[n]$ with exactly $x$ elements for any integer $x$ between $g_{k,b,q} (n)$ and $f_{k,b,q} (n)$, inclusively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Journal of Combinatorial Mathematics and Combinatorial Computing 120, no. 1 (2024) 315-21",
  "arxiv": "2610.17188",
  "author_total": 1,
  "authors": [
    {
      "name": "Minh Quang Nguyên"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vbkh7xhqb2wm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17188",
  "fulltext_url": null,
  "id": "2vbkh7xhqb2wm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17188",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221792311",
  "pdf_url": "https://arxiv.org/pdf/2610.17188",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vbkh7xhqb2wm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17188",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17188"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Generalisation of Maximal $(k,b)$-Linear-Free Sets of Integers",
  "title_text": "A Generalisation of Maximal (k,b)-Linear-Free Sets of Integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/2vbkh7xhqb2wm",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.61091/jcmcc120-28",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}