 Chromatic number of Euclidean space

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 $\chi ({\mathbb{R}}^{4p-1})\ge \left(\genfrac{}{}{0ex}{}{4p-1}{2p-1}\right)/\left(\genfrac{}{}{0ex}{}{4p-1}{p-1}\right)$.