Obstructions to coloring arithmetic graphs

Source abstract

The arithmetic graph $B_n$ joins distinct $a,b\in\N$ when $\max(a,b)/\gcd(a,b)\le n$. We prove $χ(B_{205})=206$, disproving the conjecture that $χ(B_n)=n$ for every $n$, equivalently the Rainbow Cascades Conjecture. The proof reduces an arbitrary tiling by the arithmetic exponent tile to a periodic tiling, then to two families of finite quotients, which are excluded using exact computations. We also construct a $208$-coloring using $\Z_{104}\times\Z_2$ and prove $212\leχ(B_{211})\le213$. The lower bound at $211$ follows from prime-cardinality tiling rigidity and the published nonexistence of a cyclic logarithm of length $211$; we give a direct proof of the required rigidity statement. Finally, we record the equivalence with the List Cascade Coloring Conjecture and the conjecture on ironic decorations, and deduce finite graph counterexamples to both. The least $n$ with $χ(B_n)>n$ is either $195$ or $205$; determining which remains open.