Unambiguous DNFs and Alon-Saks-Seymour

Kaspars Balodis, Shalev Ben-David, Mika Göös, Siddhartha Jain, Robin Kothari

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.

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