Unambiguous DNFs and Alon-Saks-Seymour

Abstract

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.

v1: 12 pages, 2 figures. v2: Added an author; improved result; 15 pages