{
  "abstract": "Following their resolution of the Erdős $B+B+t$ problem, Kra, Moreira, Richter, and Robertson posed a number of questions and conjectures related to infinite configurations in positive density subsets of the integers and other amenable groups. We give a negative answer to several of their questions and conjectures by producing families of counterexamples based on a construction of Ernst Straus.Included among our counterexamples, we exhibit, for any $\\varepsilon > 0$, a set $A \\subseteq \\mathbb{N}$ with multiplicative upper Banach density at least $1 - \\varepsilon$ such that $A$ does not contain any dilated product set $\\{b_1b_2t : b_1, b_2 \\in B, b_1 \\ne b_2\\}$ for an infinite set $B \\subseteq \\mathbb{N}$ and $t \\in \\mathbb{Q}_{>0}$. We also prove the existence of a set $A \\subseteq \\mathbb{N}$ with additive upper Banach density at least $1 - \\varepsilon$ such that $A$ does not contain any polynomial configuration $\\{b_1^2 + b_2 + t : b_1, b_2 \\in B, b_1 < b_2\\}$ for an infinite set $B \\subseteq \\mathbb{N}$ and $t \\in \\mathbb{Z}$. Counterexamples to some closely related problems are also discussed.",
  "abstract_shown": true,
  "affiliations": [
    "EPFL"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Ethan Ackelsberg",
      "orcid": "0000-0002-0063-8229"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftwb407g5prws.bib",
  "cited_by": 12,
  "doi": "10.37236/14787",
  "fulltext_url": null,
  "id": "ftwb407g5prws",
  "issns": [
    "1077-8926",
    "1097-1440"
  ],
  "landing_url": "https://doi.org/10.37236/14787",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W4396243886",
  "pdf_url": "https://www.combinatorics.org/ojs/index.php/eljc/article/download/v33i3p89/pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": "Electronic Journal of Combinatorics",
  "ris": "https://research.draconitic.com/paper/ftwb407g5prws.ris",
  "sources": [
    {
      "first_seen": "2026-11-07T13:20:42Z",
      "native_id": "W4396243886",
      "source": "openalex",
      "url": "https://doi.org/10.37236/14787"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Limits and Structures in Graph Theory",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counterexamples to Generalizations of the Erdős $B+B+t$ Problem",
  "title_text": "Counterexamples to Generalizations of the Erdős B+B+t Problem",
  "type": "peer-reviewed",
  "updated_at": "2026-11-08T13:20:36Z",
  "url": "https://research.draconitic.com/paper/ftwb407g5prws",
  "venue": "The Electronic Journal of Combinatorics",
  "versions": []
}