# Several Separations Based on a Partial Boolean Function

We show a partial Boolean function f together with an input x∈ f^-1(*) such that both C_0̅(f,x) and C_1̅(f,x) are at least C(f)^2-o(1). Due to recent results by Ben-David, Göös, Jain, and Kothari, this result implies several other separations in query and communication complexity. For example, it gives a function f with C(f)=Ω(deg^2-o(1)(f)) where C and deg denote certificate complexity and polynomial degree of f. (This is the first improvement over a separation between C(f) and deg(f) by Kushilevitz and Nisan in 1995.) Other implications of this result are an improved separation between sensitivity and polynomial degree, a near-optimal lower bound on conondeterministic communication complexity for Clique vs. Independent Set problem and a near-optimal lower bound on complexity of Alon–Saks–Seymour problem in graph theory.

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