Measurable obstructions for unmeasurable colourings

Source abstract

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 $\overlineα(\mathbb{R}^4)=m_1(\mathbb{R}^4)$ and $χ(\mathbb{R}^4)=χ^{(m)}(\mathbb{R}^4)$, and for $d\ge5$ that $$\overlineα(\mathbb{Q}^d) = \overlineα(\mathbb{R}^d)=m_1(\mathbb{R}^d) \qquad\text{and}\qquad χ(\mathbb{Q}^d) = χ(\mathbb{R}^d)=χ^{(m)}(\mathbb{R}^d).$$ 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 $$\sup_{|D|=k}χ_D(\mathbb{R}^d) \ge m_1(\mathbb{R}^d)^{-k}.$$ Making progress on another problem of Erdős, we prove that $$(2+o(1))^d \le χ(\mathbb{R}^d) \le \left(\frac{3\sqrt{3}}{4}+o(1)\right)^d.$$ We also vastly improve the lower bounds for $χ(\mathbb{R}^d)$ for small $d\ge4$. We expect that our techniques could be developed much further. This includes the possibility of extending our main theorem that $χ(\mathbb{R}^d)=χ^{(m)}(\mathbb{R}^d)$ for $d\ge 4$ to $d=3$ or possibly even to $d=2$ to tackle the Hadwiger-Nelson problem.