Unambiguous DNFs and Alon-Saks-Seymour

02/16/2021

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by   Kaspars Balodis, et al.

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We exhibit an unambiguous k-DNF formula that requires CNF width Ω̃(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.