= Chromatic number of Euclidean space
{title2=$\chi(\mathbb R^n)$}
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 $\binom{4p-1}{p-1}$, and hence gives $\chi(\mathbb R^{4p-1})\geq\binom{4p-1}{2p-1}/\binom{4p-1}{p-1}$.
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