It was known that if the treewidth of the core of the structure
$A$ is less than $k$, then CSP($A$,-) is solvable by the
$k$-consistency algorithm. We prove the exact converse to
this: if the treewidth of the core of $A$ is at least $k$, then
CSP($A$,-) is not solvable by the $k$-consistency algorithm.
To build our counterexample, we use the characterization
of treewidth in terms of brambles.
This reports joint work with Andrei Bulatov and Victor Dalmau,