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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
Sharp thresholds of graph properties, and the -sat problem
Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone
Boolean Functions With Low Average Sensitivity Depend On Few Coordinates
TLDR
It is shown here that if the average sensitivity of is then can be approximated by a function depending on coordinates where is a constant depending only on the accuracy of the approximation but not on .
On the measure of intersecting families, uniqueness and stability
  • E. Friedgut
  • Mathematics, Computer Science
    Comb.
  • 1 September 2008
TLDR
It is proved, for a certain range of parameters, that the t-intersecting families of maximal measure are the families of all sets containing t fixed elements, and that the extremal examples are not only unique, but also stable.
Intersecting Families of Permutations
A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently
Intersecting Families are Essentially Contained in Juntas
  • Irit Dinur, E. Friedgut
  • Computer Science, Mathematics
    Combinatorics, Probability and Computing
  • 1 March 2009
TLDR
It is proved that every intersecting family of k-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersectingfamily), and the methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.
A sharp threshold for k-colorability
. ABSTRACT: Let k be a fixed integer and fn , p denote the probability that the random k . . graph Gn , p is k-colorable. We show that for k G 3, there exists dn such that for any k e ) 0, dn y e
Graph Products, Fourier Analysis and Spectral Techniques
Abstract.We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among
On the number of copies of one hypergraph in another
AbstractGiven two hypergraphsH andG, letN(G, H) denote the number of subhypergraphs ofG isomorphic toH. LetN(l, H) denote the maximum ofN(G, H), taken over allG with exactlyl edges. In [1] Noga Alon
Boolean functions whose Fourier transform is concentrated on the first two levels
In this note we describe Boolean functions f(x1,x2,?,xn) whose Fourier coefficients are concentrated on the lowest two levels. We show that such a function is close to a constant function or to a
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