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

Unambiguous DNFs and Alon-Saks-Seymour