Chromatic number of Euclidean space ($\chi ({\mathbb{R}}^n)$)

This is the least number of colors needed to color all points of Euclidean space so that points at Euclidean distance one receive different colors. Any finite unit-distance graph embedded in the space gives a lower bound. If $p$ is prime, the characteristic vectors of all $(2p-1)$-subsets of $[4p-1]$, divided by $\sqrt{2p}$, give such a graph: its edges are exactly pairs intersecting in $p-1$ points. The prime-power modular intersection bound bounds an independent set by $\left(\genfrac{}{}{0ex}{}{4p-1}{p-1}\right)$, and hence gives $\chi ({\mathbb{R}}^{4p-1})\ge \left(\genfrac{}{}{0ex}{}{4p-1}{2p-1}\right)/\left(\genfrac{}{}{0ex}{}{4p-1}{p-1}\right)$.