Unambiguous DNFs and Alon-Saks-Seymour

We exhibit an unambiguous
$k$
-DNF formula that requires CNF width
$\tilde\Omega(k^{2})$
, which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon–Saks–Seymour problem in graph theory (posed in 1991), which asks: How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query and communication complexity.