Erdős Problem #706

Let \(L(r)\) be such that if \(G\) is a graph formed by taking a finite set of points \(P\) in \(\mathbb{R}^2\) and some set \(A\subset (0,\infty)\) of size \(r\), where the vertex set is \(P\) and there is an edge between two points if and only if their distance is a member of \(A\), then \(\chi(G)\leq L(r)\). Estimate \(L(r)\). In particular, is it true that \(L(r)\leq r^{O(1)}\)? The case \(r=1\) is the Hadwiger-Nelson problem, for which it is known that \(5\leq L(1)\leq 7\). See also [508](https://www.erdosproblems.com/508), [704](https://www.erdosproblems.com/704), and [705](https://www.erdosproblems.com/705). Here \(L(r)\leq r^{O(1)}\) is read as: there is \(C\) with \(\chi(G)\leq r^C\) for all large \(r\). Small \(r\) must be excluded, since \(L(1)\geq 5 > 1^C\).