The exact region between Chatterjee's $ξ$ and Blomqvist's $β$
We determine the exact region of Chatterjee's rank correlation $ξ$ and Blomqvist's $β$, that is, the set of all pairs $(ξ(C),β(C))$ where $C$ ranges over all bivariate copulas. The region equals $\{(x,y)\in[0,1]\times[-1,1]:|y|^3\le2x\}$, so every copula satisfies the sharp inequality $|β(C)|^3\le2ξ(C)$. Its left boundary $ξ=|β|^3/2$ is attained uniquely by an explicit family of copulas that perturb independence by a signed tent function centered at the median, and its right boundary $ξ=1$ by deterministic interval-exchange copulas. The proof reduces the problem to a one-dimensional minimization of an $L^2$-norm over $1$-Lipschitz functions. We further show that the region does not shrink when restricted to radially symmetric copulas and that positively quadrant dependent copulas attain exactly its upper half. For copulas under which the second variable is stochastically increasing in the first, the exact region is $\{(x,y)\in[0,1]^2:y^3\le2x,\ 3(1-y)^2\le4(1-x)\}$, and its right boundary is attained uniquely by explicit singular two-branch copulas. In particular, under stochastic increasingness $β=0$ is compatible with every value $ξ\in[0,1/4]$, whereas $ξ>1/4$ forces $β>0$.