A Generalized Turan Problem and its Applications

Our first theorem is a hierarchy theorem for the query complexity of testing graph properties with one-sided error; more precisely, we show that for every sufficiently fast-growing function f from (0,1) to the natural numbers, there is a graph property whose one-sided-error query complexity is precisely f(\Theta(\epsilon)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is exponential in 1/\epsilon. Our hierarchy theorem partially resolves this problem by exhibiting a property whose onesided-error query complexity is exponential in 1/\epsilon. We also use our hierarchy theorem in order to resolve a problem raised by Alon and Shapira [STOC 2005] regarding testing relaxed versions of bipartiteness.

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