{
  "abstract": "Let $T_q(n,r)$ be the largest size of a vertex set inducing a triangle-free subgraph of the graph on $[q]^n$ whose edges join words at Hamming distance $r$. We study this quantity for fixed alphabet size and fixed distance, with particular emphasis on the ternary cube. Alphabet restriction transfers the binary fixed-distance theorem of Balogh, Chen, and Li to give $T_q(n,2h)=O_h(q^n/n^h)$. We extend the known distance-one formula and distance-two order to arbitrary integer alphabet sizes, obtaining $T_q(n,1)=2q^{n-1}$ and $T_q(n,2)=Θ_q(q^n/n)$, and establish $T_3(n,4)=Θ(3^n/n^2)$. For general fixed distance, we prove \\[ T_q(n,r)=Ω_{q,r}\\!\\left( \\frac{q^n\\sqrt{\\log n}}{n^{(r+\\lfloor r/2\\rfloor)/2}} \\right)\\qquad(q\\ge3), \\] while we show that $T_q(n,r)=Ω_{q,r}(q^n/n^{\\lfloor(q-1)r/q\\rfloor})$ holds for prime-power $q$ and $r\\ge2$ via the extended BCH parity-check constructions. At distance three, we construct coordinate weights avoiding bounded-coefficient three-term relations and obtain \\[ T_q(n,3)\\ge q^n/[n\\exp(O_q(\\sqrt{\\log n}))]. \\] Our main ternary construction combines cap partitions, multicolor triangle Ramsey colorings, and a quadratic map to obtain \\[ T_3(n,3)=Ω\\!\\left( \\frac{\\log\\log n}{\\log n}\\,3^n\\right). \\] A coordinate-line counting argument gives the complementary upper bound $T_3(n,3)\\le(5/27+O(n^{-1}))3^n$, together with an explicit bound for every fixed $q\\ge3$. Whether the ternary density tends to zero remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32789",
  "author_total": 3,
  "authors": [
    {
      "name": "Bence Csajbók"
    },
    {
      "name": "Padmini Mukkamala"
    },
    {
      "name": "Zoltán Lóránt Nagy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c1306fb259ht7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32789",
  "fulltext_url": null,
  "id": "c1306fb259ht7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32789",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223686683",
  "pdf_url": "https://arxiv.org/pdf/2610.32789",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c1306fb259ht7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32789",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32789"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Triangle-Free Sets in Exact-Distance Hamming Graphs",
  "title_text": "Triangle-Free Sets in Exact-Distance Hamming Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/c1306fb259ht7",
  "venue": "arXiv",
  "versions": []
}