We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

/pdf/the-probabilistic-method-5g63k0itke.pdf

The Probabilistic Method

We study the problem of searching on data that is encrypted using a public key system. Consider user Bob who sends email to user Alice encrypted under Alice’s public key. An email gateway wants to test whether the email contains the keyword “urgent” so that it could route the email accordingly. Alice, on the other hand does not wish to give the gateway the ability to decrypt all her messages. We define and construct a mechanism that enables Alice to provide a key to the gateway that enables the gateway to test whether the word “urgent” is a keyword in the email without learning anything else about the email. We refer to this mechanism as Public Key Encryption with keyword Search. As another example, consider a mail server that stores various messages publicly encrypted for Alice by others. Using our mechanism Alice can send the mail server a key that will enable the server to identify all messages containing some specific keyword, but learn nothing else. We define the concept of public key encryption with keyword search and give several constructions.

https://link.springer.com/content/pdf/10.1007%2F978-3-540-24676-3_30.pdf

Public Key Encryption with Keyword Search

This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of these methods sprung off from the development of geometric functional analysis since the 1970's. They have applications in several fields, most notably in theoretical computer science, statistics and signal processing. A few basic applications are covered in this text, particularly for the problem of estimating covariance matrices in statistics and for validating probabilistic constructions of measurement matrices in compressed sensing. These notes are written particularly for graduate students and beginning researchers in different areas, including functional analysts, probabilists, theoretical statisticians, electrical engineers, and theoretical computer scientists.

Introduction to the non-asymptotic analysis of random matrices.

Concentration functions and inequalities Isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index

/pdf/the-concentration-of-measure-phenomenon-27qyoor2ef.pdf

The concentration of measure phenomenon

LetA=(a ij ) be ann ×n matrix whose entries fori≧j are independent random variables anda ji =a ij . Suppose that everya ij is bounded and for everyi>j we haveEa ij =μ,D 2 a ij =σ2 andEa ii =v.

/pdf/the-eigenvalues-of-random-symmetric-matrices-i0akvx5jld.pdf

The eigenvalues of random symmetric matrices

Letfr(n, k) denote the maximum number ofk-subsets of ann-set satisfying the condition in the title. It is proved that 

$$f_1 (n,r(t - 1) + 1 + d)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } (_{ t}^{n - d} )/(_{ t}^{k - d} )$$

wheneverd=0, 1 ord≦r/2t2 with equality holding iff there exists a Steiner systemS(t, r(t−1)+1,n−d). The determination offr(n, 2r) led us to a new generalization of BIBD (Definition 2.4). Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family.

Families of finite sets in which no set is covered by the union ofr others

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.

The History of Degenerate (Bipartite) Extremal Graph Problems

Almost all combinatorial question can be reformulated as either a matching or a covering problem of a hypergraph. In this paper we survey some of the important results.

Zoltán Füredi

Papers

The eigenvalues of random symmetric matrices

Families of finite sets in which no set is covered by the union ofr others

The History of Degenerate (Bipartite) Extremal Graph Problems

Matchings and covers in hypergraphs

Onr-Cover-free Families