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Measurable obstructions for unmeasurable colourings — arXiv:2610.12301

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.12301
License (as archived): CC0-1.0
Archived: 2026-10-09T06:29:12.260Z
SHA-256 of the archived text: 2a9ad75c4faf6b55db8f4a59c3693f42bc0f52b1da584a3e2e81cdf0fc047e61
Stable URL of this page: https://metraipolis.com/documento/dc_aa67861bec4e

How to cite

Measurable obstructions for unmeasurable colourings — arXiv:2610.12301 — arXiv (en), https://arxiv.org/abs/2610.12301 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_aa67861bec4e (Metraipolis archive dc_aa67861bec4e, sha256 2a9ad75c4faf6b55db8f4a59c3693f42bc0f52b1da584a3e2e81cdf0fc047e61)

BibTeX

@misc{dcaa67861bec4e,
  title = {Measurable obstructions for unmeasurable colourings — arXiv:2610.12301},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.12301},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_aa67861bec4e; sha256 2a9ad75c4faf6b55db8f4a59c3693f42bc0f52b1da584a3e2e81cdf0fc047e61},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.12301. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Due to the availability of powerful analytic techniques, vastly superior lower bounds are known for the measurable chromatic number of Euclidean spaces compared to their ordinary chromatic number. Indeed, even the breakthrough lower bound of 5 for the famous Hadwiger-Nelson problem lagged over 35 years behind that of the measurable setting. This raises the fundamental question of whether the measurable and ordinary chromatic number of Euclidean spaces differ as conjectured by Székely in 1984. Our main result is that $\overline\alpha(\mathbb{R}^4)=m_1(\mathbb{R}^4)$ and $\chi(\mathbb{R}^4)=\chi^{(m)}(\mathbb{R}^4)$, and for $d\ge5$ that \[ \overline\alpha(\mathbb{Q}^d) = \overline\alpha(\mathbb{R}^d)=m_1(\mathbb{R}^d) \qquad\text{and}\qquad \chi(\mathbb{Q}^d) = \chi(\mathbb{R}^d)=\chi^{(m)}(\mathbb{R}^d). \] Our theorem also holds for multiple forbidden distances $D=\{d_1,\ldots,d_t\}$ provided that $d_1^2,\ldots,d_t^2 \in \mathbb{Q}$. As a consequence, we immediately lift numerous measurable chromatic number results into the ordinary setting. We also take the opportunity to further optimize the new bounds. For multiple distances, Erdős asked whether the chromatic number of $\mathbb{R}^d$ with up to $k$ forbidden distances $D$ grows exponentially in $k$. By a theorem of Bukh, we obtain for $d \ge 4$ that \[ \sup_{|D|=k}\chi_D(\mathbb{R}^d) \ge m_1(\mathbb{R}^d)^{-k}. \] Making progress on another problem of Erdős, we prove that \[ (2+o(1))^d \le \chi(\mathbb{R}^d) \le \left(\frac{3\sqrt{3}}{4}+o(1)\right)^d. \] We also vastly improve the lower bounds for $\chi(\mathbb{R}^d)$ for small $d\ge4$. We expect that our techniques could be developed much further. This includes the possibility of extending our main theorem that $\chi(\mathbb{R}^d)=\chi^{(m)}(\mathbb{R}^d)$ for $d\ge 4$ to $d=3$ or possibly even to $d=2$ to tackle the Hadwiger-Nelson problem.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO) ; Classical Analysis and ODEs (math.CA); Metric Geometry (math.MG).
Pagina del lavoro: https://arxiv.org/abs/2610.12301 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-09T05:53:57.301Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).