Gil Kalai

The influence of variables on Boolean functions

Methods from harmonic analysis are used to prove some general theorems on Boolean functions. These connections with harmonic analysis viewed by the authors are very promising; besides the results on Boolean functions they enable them to prove theorems on the rapid mixing of the random walk on the cube and in the extremal theory of finite sets.<<ETX>>

Every monotone graph property has a sharp threshold

In their seminal work which initiated random graph theory Erdos and Renyi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let Vn(p) = {0, 1}n denote the Hamming space endowed with the probability measure μp defined by μp( 1, 2, . . . , n) = pk · (1 − p)n−k, where k = 1 + 2 + · · · + n. Let A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on {1, 2, . . . , n} such that A is invariant under Γ. Theorem. For every symmetric monotone A, if μp(A) > then μq(A) > 1− for q = p+ c1 log(1/2 )/ logn. (c1 is an absolute constant.)

Noise Sensitivity of Boolean Functions and Applications to Percolation

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given.

Rigidity and the lower bound theorem 1

SummaryFor an arbitrary triangulated (d-1)-manifold without boundaryC withf0 vertices andf1 edges, define

Elections Can be Manipulated Often

\gamma (C) = f_1 - df_0 + \left( {\begin{array}{*{20}c} {d + 1} \\ 2 \\ \end{array} } \right)

A quasi-polynomial bound for the diameter of graphs of polyhedra

. Barnette proved that γ(C)≧0. We use the rigidity theory of frameworks and, in particular, results related to Cauchys rigidity theorem for polytopes, to give another proof for this result. We prove that ford≧4, if γ(C)=0 thenC is a triangulated sphere and is isomorphic to the boundary complex of a stacked polytope. Other results: (a) We prove a lower bound, conjectured by Björner, for the number ofk-faces of a triangulated (d-1)-manifold with specified numbers of interior vertices and boundary vertices. (b) IfC is a simply connected triangulatedd-manifold,d≧4, and γ(lk(v, C))=0 for every vertexv ofC, then γ(C)=0. (lk(v,C) is the link ofv inC.) (c) LetC be a triangulatedd-manifold,d≧3. Then Ske11(Δd+2) can be embedded in skel1 (C) iff γ(C)>0. (Δd is thed-dimensional simplex.) (d) IfP is a 2-simpliciald-polytope then

Polytopes : combinatorics and computation

f_1 (P) \geqq df_0 (P) - \left( {\begin{array}{*{20}c} {d + 1} \\ 2 \\ \end{array} } \right)