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PreprintCC BY 4.0arXiv

Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures

  • Deep Bhattacharjee
  • Priyabrata Mandal
  • Ushashi Bhattacharya
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Abstract

We prove both Erdős-Ulam conjectures on monochromatic union-closed families. Every two-colouring of the subsets of a finite set contains a monochromatic union-closed family whose size grows faster than any fixed power of the size of the set, while suitable colourings admit no monochromatic union-closed family of exponential size. Both results hold for any number of colours and are verified in Lean.

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Cite

BibTeXRIS
BibTeX
@misc{bhattacharjee2026resolving,
  title = {{Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures}},
  author = {Deep Bhattacharjee and Priyabrata Mandal and Ushashi Bhattacharya},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02833},
  eprint = {2610.02833},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02833},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures
AU  - Deep Bhattacharjee
AU  - Priyabrata Mandal
AU  - Ushashi Bhattacharya
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.02833
UR  - https://arxiv.org/abs/2610.02833
AB  - We prove both Erdős-Ulam conjectures on monochromatic union-closed families. Every two-colouring of the subsets of a finite set contains a monochromatic union-closed family whose size grows faster than any fixed power of the size of the set, while suitable colourings admit no monochromatic union-closed family of exponential size. Both results hold for any number of colours and are verified in Lean.
ER  -