KKL, Kruskal-Katona, and Monotone Nets

We generalize the Kahn-Kalai-Linial (KKL) Theorem to random walks on Cayley and Schreier graphs, making progress on an open problem of Hoory, Linial, and Wigderson. In our generalization, the underlying group need not be abelian so long as the generating set is a union of conjugacy classes. An example corollary is that for every f : [n] k → {0, 1} with E[f] and k/n bounded away from 0 and 1, there is a pair 1 ≤ i < j ≤ n such that Iij(f) ≥ Ω(log n n ). Here Iij(f) denotes the “influence” on f of swapping the ith and jth coordinates. Using this corollary we obtain a “robust” version of the Kruskal-Katona Theorem: Given a constantdensity subset A of a middle slice of the Hamming n-cube, the density of ∂A is greater by at least Ω(log n n ), unless A is noticeably correlated with a single coordinate. As an application of these results, we show that the set of functions {0, 1, x1, . . . , xn, Maj} is a (1/2 − γ)-net for the set of all n-bit monotone boolean fun...

• Type: Conference
• Year: 2009
• Where: FOCS
• Authors: Ryan O'Donnell, Karl Wimmer
• Keywords: Example Corollary, FOCS 2009, Random Walks, Schreier Graphs, Theoretical Computer Science