Nathan Linial

• Hebrew University of Jerusalem

Hypertrees are high-dimensional counterparts of graph theoretic trees. They have attracted a great deal of attention by various investigators. Here we introduce and study hyperpaths—a particular class of hypertrees which are high dimensional analogs of paths in graph theory. A d-dimensional hyperpath is a d-dimensional hypertree in which every (d-1...

A hallmark of cancer evolution is that the tumor may change its cell identity and improve its survival and fitness. Drastic change in microRNA (miRNA) composition and quantities accompany such dynamic processes. Cancer samples are composed of cells’ mixtures of varying stages of cancerous progress. Therefore, cell-specific molecular profiling repre...

The characterization of germline genetic variation affecting cancer risk, known as cancer predisposition, is fundamental to preventive and personalized medicine. Studies of genetic cancer predisposition typically identify significant genomic regions based on family-based cohorts or genome-wide association studies (GWAS). However, the results of suc...

The characterization of germline genetic variation affecting cancer risk, known as cancer predisposition, is fundamental to preventive and personalized medicine. Current attempts to detect cancer predisposition genomic regions are typically based on small-scale familial studies or genome-wide association studies (GWAS) over dedicated case-control c...

Numerous computational methods have been developed to screening the genome for candidate driver genes based on genomic data of somatic mutations in tumors. Compiling a catalog of cancer genes has profound implications for the understanding and treatment of the disease. Existing methods make many implicit and explicit assumptions about the distribut...

It is estimated that up to 10% of cancer incidents are attributed to inherited genetic alterations. Despite extensive research, there are still gaps in our understanding of genetic predisposition to cancer. It was theorized that ultra-rare variants partially account for the missing heritable component. We harness the UK BioBank dataset of ~ 500,000...

It is well known that for every even integer n, the complete graph \(K_{n}\) has a one-factorization, namely a proper edge coloring with \(n-1\) colors. Unfortunately, not much is known about the possible structure of large one-factorizations. Also, at present we have only woefully few explicit constructions of large one-factorizations. In particul...

We introduce Proteome-Wide Association Study (PWAS), a new method for detecting gene-phenotype associations mediated by protein function alterations. PWAS aggregates the signal of all variants jointly affecting a protein-coding gene and assesses their overall impact on the protein's function using machine learning and probabilistic models. Subseque...

Mature microRNAs (miRNAs) are small, non-coding RNA molecules that function by base-pairing with mRNAs. In multicellular organisms, miRNAs lead to mRNA destabilization and translation arrest. Importantly, the quantities and stichometry of miRNAs/mRNAs determine the miRNA regulation characteristics of specific cells. In this study, we used COMICS (C...

Over the last two decades, Genome-Wide Association Study (GWAS) has become a canonical tool for exploratory genetic research, generating countless gene-phenotype associations. Despite its accomplishments, several limitations and drawbacks still hinder its success, including low statistical power and obscurity about the causality of implicated varia...

It is estimated that up to 10% of cancer incidents are attributed to inherited genetic alterations. Despite extensive research, there are still gaps in our understanding of genetic predisposition to cancer. It was theorized that ultra-rare variants partially account for the missing heritable component. We harness the UK BioBank dataset of ~500,000...

An internal partition of a graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E)$$\end{document} is a partitioning of V into two parts such that every ve...

Mature microRNAs (miRNAs) regulate most human genes through direct base-pairing with mRNAs. We investigate the underlying principles of miRNA regulation in living cells. To this end, we overexpressed miRNAs in different cell types and measured the mRNA decay rate under a paradigm of a transcriptional arrest. Based on an exhaustive matrix of mRNA-mi...

Over the last two decades, GWAS (Genome-Wide Association Study) has become a canonical tool for exploratory genetic research, generating countless gene-phenotype associations. Despite its accomplishments, several limitations and drawbacks still hinder its success, including low statistical power and obscurity about the causality of implicated varia...

Compiling the catalogue of genes actively involved in cancer is an ongoing endeavor, with profound implications to the understanding and treatment of the disease. An abundance of computational methods have been developed to screening the genome for candidate driver genes based on genomic data of somatic mutations in tumors. Existing methods make ma...

Mature microRNAs (miRNAs) regulate most human genes through direct base-pairing with mRNAs. We investigate some underlying principles of such regulation. To this end, we overexpressed miRNAs in different cell types and measured the mRNA decay rate under transcriptional arrest. Parameters extracted from these experiments were incorporated into a com...

Compiling the catalogue of genes actively involved in tumorigenesis (known as cancer drivers) is an ongoing endeavor, with profound implications to the understanding of tumorigenesis and treatment of the disease. An abundance of computational methods have been developed to screening the genome for candidate driver genes based on genomic data of som...

An internal partition of a graph is a partitioning of the vertex set into two parts such that for every vertex, at least half of its neighbors are on its side. We prove that for every positive integer $r$, asymptotically almost every $2r$-regular graph has an internal partition.

This article surveys some of the work done in recent years on random simplicial complexes. We mostly consider higher-dimensional analogs of the well known phase transition in G(n, p) theory that occurs at \(p = \frac{1} {n}\). Our main objective is to provide a more streamlined and unified perspective of some of the papers in this area.

The primary function of microRNAs (miRNAs) is to maintain cell homeostasis. In cancerous tissues miRNAs' expression undergo drastic alterations. In this study, we use miRNA expression profiles from The Cancer Genome Atlas of 24 cancer types and 3 healthy tissues, collected from >8500 samples. We seek to classify the cancer's origin and tissue ident...

The primary function of microRNAs (miRNAs) is to maintain cell homeostasis. In cancerous tissues miRNAs’ expression undergo drastic alterations. In this study, we used miRNA expression profiles from The Cancer Genome Atlas (TCGA) of 24 cancer types and 3 healthy tissues, collected from >8500 samples. We seek to classify the cancer’s origin and tiss...

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erd\H{o}s-Szekeres theorem: For every $k\ge1$, every order-$n$ $k$-dimensional permutation contains a monotone subsequence of length $\Omega_{k}\left(\sqrt{n}\right)$, and this is tight. On the other hand, and unlike the classical case, the...

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erd\H{o}s-Szekeres theorem: For every $k\ge1$, every order-$n$ $k$-dimensional permutation contains a monotone subsequence of length $\Omega_{k}\left(\sqrt{n}\right)$, and this is tight. On the other hand, and unlike the classical case, the...

Let $L$ be an order-$n$ Latin square. For $X, Y, Z \subseteq \{1, ... ,n\}$, let $L(X, Y. Z)$ be the number of triples $i\in X, j\in Y, k\in Z$ such that $L(i,j) = k$. We conjecture that asymptotically almost every Latin square satisfies $|L(X, Y, Z) - \frac 1n |X||Y||Z||\le O(\sqrt{|X||Y||Z|})$ for every $X, Y$ and $Z$. Let $\varepsilon(L):= \max...

Let $L$ be an order-$n$ Latin square. For $X, Y, Z \subseteq \{1, ... ,n\}$, let $L(X, Y. Z)$ be the number of triples $i\in X, j\in Y, k\in Z$ such that $L(i,j) = k$. We conjecture that asymptotically almost every Latin square satisfies $|L(X, Y, Z) - \frac 1n |X||Y||Z||\le O(\sqrt{|X||Y||Z|})$ for every $X, Y$ and $Z$. Let $\varepsilon(L):= \max...

Several years ago Linial and Meshulam introduced a model called X_d(n,p) of random n-vertex d-dimensional simplicial complexes. The following question suggests itself very naturally: What is the threshold probability p=p(n) at which the d-dimensional homology of such a random d-complex is, almost surely, nonzero? Here we derive an upper bound on th...

It is well-known that the $G(n,p)$ model of random graphs undergoes a dramatic change around $p=\frac 1n$. It is here that the random graph is, almost surely, no longer a forest, and here it first acquires a giant (i.e., order $\Omega(n)$) connected component. Several years ago, Linial and Meshulam have introduced the $X_d(n,p)$ model, a probabilit...

Motivation: Modern protein sequencing techniques have led to the determination of >50 million protein sequences. ProtoNet is a clustering system that provides a continuous hierarchical agglomerative clustering tree for all proteins. While ProtoNet performs unsupervised classification of all included proteins, finding an optimal level of granularity...

What is the higher-dimensional analog of a permutation? If we think of a permutation as given by a permutation matrix, then the following definition suggests itself: A d-dimensional permutation of order n is an n×n×..×n=[n]d+1 array of zeros and ones in which every line contains a unique 1 entry. A line here is a set of entries of the form {(x1,..,...

We give a very simple proof of a strengthened version of Chernoff's Inequality. We derive the same conclusion from much weaker assumptions.

Richard Wilson conjectured in 1974 the following asymptotic formula for the number of n ‐vertex Steiner triple systems: \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*} STS(n) = \left( (1+o(1))\frac{n}{e^2} \right)^{\frac{n^2}{6}}\end{align*}\end{document} . Our main result is that The pr...

We give lower bounds on the maximum possible girth of an r-uniform, d-regular hypergraph with at most n vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between 3/2+o(1) and 2+o(1)). We also define a random r-uniform ’Cayley’ hypergraph...

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k−1)-dimensional skeleton and \binomn-1k\binom{n-1}{k} facets such that H k (X;ℚ)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers with k+1 and n relatively prime, and let A be a (k+1)-element subset of the cyclic group ℤ n . T...

Let us denote by Ω n the Birkhoff polytope of n×n doubly stochastic matrices. As the Birkhoff–von Neumann theorem famously states, the vertex set of Ω n coincides with the set of all n×n permutation matrices. Here we consider a higher-dimensional analog of this basic fact. Let \(\varOmega^{(2)}_{n}\) be the polytope which consists of all tristochas...

In this article, we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G1 and G2, if V(H) = V(G1) × V(G2) and both projection maps V(H)→V(G1) and V(H)→V(G2) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP-hard to decid...

ProtoNet 6.0 (http://www.protonet.cs.huji.ac.il) is a data structure of protein families that cover the protein sequence space. These families are generated through an unsupervised bottom–up clustering algorithm. This algorithm organizes large sets of proteins in a hierarchical tree that yields high-quality protein families. The 2012 ProtoNet (Vers...

Motivation: Much of the large-scale molecular data from living cells can be represented in terms of networks. Such networks occupy a central position in cellular systems biology. In the protein–protein interaction (PPI) network, nodes represent proteins and edges represent connections between them, based on experimental evidence. As PPI networks ar...

What is the higher-dimensional analog of a permutation? If we think of a permutation as given by a permutation matrix, then the following definition suggests itself: A d-dimensional permutation of order n is an n×n×...×n=[n]d+1 array of zeros and ones in which every line contains a unique 1 entry. A line here is a set of entries of the form {(x 1,....

Large-scale data collection technologies have come to play a central role in biological and biomedical research in the last decade. Consequently, it has become a major goal of functional genomics to develop, based on such data, a comprehensive description of the functions and interactions of all genes and proteins in a genome. Most large-scale biol...

Motivation: Large-scale RNA expression measurements are generating enormous quantities of data. During the last two decades, many methods were developed for extracting insights regarding the interrelationships between genes from such data. The mathematical and computational perspectives that underlie these methods are usually algebraic or probabil...

To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g n denote the expected genus of a randomly chosen oriented c...

We view the RSK correspondence as associating to each permutation $\pi \in S_n$ a Young diagram $\lambda=\lambda(\pi)$, i.e. a partition of $n$. Suppose now that $\pi$ is left-multiplied by $t$ transpositions, what is the largest number of cells in $\lambda$ that can change as a result? It is natural refer to this question as the search for the Lip...

Let Y be a random d-dimensional subcomplex of the (n-1)-dimensional simplex S obtained by starting with the full (d-1)-dimensional skeleton of S and then adding each d-simplex independently with probability p=c/n. We compute an explicit constant gamma_d=Theta(log d) so that for c < gamma_d such a random simplicial complex either collapses to a (d-1...

In this paper we study a new product of graphs called {\em tight product}. A graph $H$ is said to be a tight product of two (undirected multi) graphs $G_1$ and $G_2$, if $V(H)=V(G_1)\times V(G_2)$ and both projection maps $V(H)\to V(G_1)$ and $V(H)\to V(G_2)$ are covering maps. It is not a priori clear when two given graphs have a tight product (in...

Let Y be a random d-dimensional subcomplex of the (n-1)-dimensional simplex S obtained by starting with the full (d-1)-dimensional skeleton of S and then adding each d-simplex independently with probability p=c/n. Let c_d < d+1 be the unique positive solution of the equation (d+1)(x+1)e^{-x}+x(1-e^{-x})^{d+1}=d+1. It is shown that if c>c_d then a.a...

Age-dependent codon usage for human genes. This file depicts the age-dependent codon usage for human genes, for each of the 18 degenerately coded amino acids (denoted by one letter codes).

Age-dependent codon usage for fly genes. This file portrays the age-dependent codon usage for fly genes, for each of the 18 degenerately coded amino acids.

Age-dependent codon usage for mouse genes. This file shows the age-dependent codon usage for mouse genes, for each of the 18 degenerately coded amino acids.

Codon usage may vary significantly between different organisms and between genes within the same organism. Several evolutionary processes have been postulated to be the predominant determinants of codon usage: selection, mutation, and genetic drift. However, the relative contribution of each of these factors in different species remains debatable....

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP--hard problems are easier to solve. In particular, whether there exist algorithms that solve correctly and in polyno...

To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus of a randomly chosen oriented c...

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k-1)-dimensional skeleton and \binom{n-1}{k} facets such that H_k(X;Q)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers with k+1 and n relatively prime, and let A be a (k+1)-element subset of the cyclic group Z_n. The sum complex...

This paper has two main focal points. We first consider an important class of machine learning algorithms: large margin classifiers, such as Support Vector Machines. The notion of margin complexity quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative constan...

We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our main focus in this paper is on the nodal domains associated with the different eigenfunctions. In the analogo...

For a graph G and an integer t we let mcc t ( G ) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let be any non-trivial minor-closed family of graphs. We show that mcc 2 ( G ) = O ( n 2/3 ) for any n -vertex graph G ∈ . This bound is asympt...

For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that mcc2(G)=O(n2/3) for any n-vertex graph G∈F. This bound is asymptotically optimal...

Protein domains are subunits of proteins that recur throughout the protein world. There are many definitions attempting to capture the essence of a protein domain, and several systems that identify protein domains and classify them into families. EVEREST, recently described in Portugaly et al. (2006) BMC Bioinformatics, 7, 277, is one such system t...

Fréchet’s classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Fréchet embedding is Bourgain's embedding [4]. The authors have recently shown [2] that for every ε>0, anyn-point metric space contains a subset of size at leastn 1−ε which embeds into ℓ2 with distortion O(\f...

We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map � : H ! G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H h...

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields. But, perhaps, we should start with a few words about graphs in general. They are, of course, one of...

We study here lifts and random lifts of graphs, as defined by A. Amit and N. Linial [Combinatorica 22, No. 1, 1–18 (2002; Zbl 0996.05105)]. We consider the Hadwiger number η and the Hajós number σ of ℓ-lifts of K n and analyze their extremal as well as their typical values (that is, for random lifts). When ℓ=2, we show that n 2≤η≤n, and random lift...

In this note we determine exactly the expansion rate of an infinite 4-regular expander graph which is a variant of an expander due to Margulis. The vertex set of this graph consists of all points in the plane. The point (x,y) is adjacent to the points S(x,y),S 1(x,y),T(x,y),T 1(x,y) where S(x,y) = (x,x+y) and T(x,y) = (x+y,y). We show that the expa...

Background. DNA amplifications and deletions characterize cancer genome and are often related to disease evolution. Microarray based techniques for measuring these DNA copy-number changes use fluorescence ratios at arrayed DNA elements (BACs, cDNA or oligonucleotides) to provide signals at high resolution, in terms of genomic locations. These data...

Proteins are comprised of one or several building blocks, known as domains. Such domains can be classified into families according to their evolutionary origin. Whereas sequencing technologies have advanced immensely in recent years, there are no matching computational methodologies for large-scale determination of protein domains and their boundar...

Two sets are non-crossing if they are disjoint or one contains the other. The non-crossing graph NC n is the graph whose vertex set is the set of nonempty subsets of [n]={1,⋯,n} with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence n...

Associated with every graph G of chromatic number χ is another graph G′. The vertex set of G′ consists of all χ-colorings of G, and two χ-colorings are adjacent when they differ on exactly one vertex. According to a conjecture of Björner and Lovász, this graph G′ must be disconnected. In this note we give a counterexample to this conjecture.

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky¿s theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper...

We show that there exist k-neighborly centrally symmetric d-dimensional polytopes with 2(n + d) vertices, where $$k(d,n)=\Theta\left(\frac{d}{1+\log ((d+n)/d)}\right).$$ We also show that this bound is tight.

We study random lifts of a graph G as defined in [1]. We prove a 0-1 law which states that for every graph G either almost every lift of G has a perfect matching, or almost none of its lifts has a perfect matching. We provide a precise description of this dichotomy. Roughly speaking, the a.s. existence of a perfect matching in the lift depends on t...

ProtoNet is an automatic hierarchical classification of the protein sequence space. In 2004, the ProtoNet (version 4.0) presents the analysis of over one million proteins merged from SwissProt and TrEMBL databases. In addition to rich visualization and analysis tools to navigate the clustering hierarchy, we incorporated several improvements that al...

A signing of a graph $G=(V,E)$ is a function $s:E \rightarrow \{-1,1\}$. A signing defines a graph $\widehat{G}$, called a {\em 2-lift of $G$}, with vertex set $V(G)\times\{-1,1\}$. The vertices $(u,x)$ and $(v,y)$ are adjacent iff $(u,v) \in E(G)$, and $x \cdot y = s(u,v)$. The corresponding signed adjacency matrix $A_{G,s}$ is a symmetric $\{-1,0...

Alignment of protein structures is a fundamental task in computational molecular biology. Good structural alignments can help detect distant evolutionary relationships that are hard or impossible to discern from protein sequences alone. Here, we study the structural alignment problem as a family of optimization problems and develop an approximate p...

We study here lifts and random lifts of graphs, as defined in [1]. We consider the Hadwiger number # and the Hajos number # of #- lifts of K , and analyze their extremal as well as their typical values (that is, for random lifts). When # = 2, we show that n, and random lifts achieve the lower bound (as n # #).

Frechet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Frechet embedding is Bourgain's embedding. The authors have recently shown that for every e>0 any n-point metric space contains a subset of size at least n^(1-e) which embeds into l_2 with distortion O(\log(2/...

The existence of small d-regular graphs of a prescribed girth g is equivalent to the existence of certain codes in the d-regular infinite tree. We show that in the tree “perfect” codes exist, but those are usually not “graphic”. We also give an explicit coloring that is “nearly perfect” as well as “nearly graphic”.

The classical Ramsey theorem, states that every graph contains either a large clique or a large independent set. Here we investigate similar dichotomic phenomena in the context of finite metric spaces. Namely, we prove statements of the form "Every finite metric space contains a large subspace that is nearly quilateral or far from being equilateral...

Associated with every graph $G$ of chromatic number $\chi$ is another graph $G'$. The vertex set of $G'$ consists of all $\chi$-colorings of $G$, and two $\chi$-colorings are adjacent when they differ on exactly one vertex. According to a conjecture of Bj\"{o}rner and Lov\'asz, this graph $G'$ must be disconnected. In this note we give a counterexa...

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper...

Associated with every graph G of chromatic number is another graph G . The vertex set of G consists of all -colorings of G, and two -colorings are adjacent when they dier on exactly one vertex. According to a conjecture of Lovasz, this graph G must be disconnected. In this note we give a counterexample to this conjecture. In this paper we refute a...

In this note we show that every n-point ultrametric embeds with constant distortion p for every 1. More precisely, we consider a special type of ultrametric with hierarchical structure called a k-hierarchically well-separated tree (k-HST). We show that any k-HST can be embedded with distortion at most 1 + O(1/k) in # O(k log n) p . These facts have...

Permanents, unlike determinants, cannot be calculated in polynomial time. The authors present a deterministic, strongly polynomial algorithm which approximates the permanent of a nonnegative n×n matrix to within a multiplicative factor of e n . To this end they develop the first strongly polynomial-time algorithm for matrix scaling – an important n...

In the one-round Voronoi game, the first player chooses an n-point set W in a square Q, and then the second player places another n-point set B into Q. The payoff for the second player is the fraction of the area of Q occupied by the regions of the points of B in the Voronoi diagram of W \cup B. We give a (randomized) strategy for the second player...

A set L of linear polynomials in variables X 1 ,X 2 ,⋯,X n with real coefficients is said to be an essential cover of the cube {0,1} n if (E1) for each v∈{0,1} n , there is a p∈L such that p(v)=0; (E2) no proper subset of L satisfies (E1), that is, for every p∈L, there is a v∈{0,1} n such that p alone takes the value 0 on v; (E3) every variable app...

The extensive study of metric spaces and their embeddings has so far focused on embeddings that preserve pairwise distances. A very intriguing concept introduced by Feige allows us to quantify the extent to which larger structures are preserved by a given embedding. We investigate this concept, focusing on several major graph families such as paths...

This paper deals with Ramsey-type theorems for metric spaces. Such a theorem states that every n point metric space contains a large subspace which can be embedded with some fixed distortion in a metric space from some special class. Our main theorem states that for any ε > 0, every n point metric space contains a subspace of size at least n1-εwhic...

Let Δn−1 denote the (n − 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of Δn−1 obtained by starting with the full 1-dimensional skeleton of Δn−1 and then adding each 2−simplex independently with probability p. Let \( H_{1} {\left( {Y;{\Bbb F}_{2} } \right)} \) denote the first homology group of Y with mod 2 coefficients. It is...

The classical Ramsey theorem states that every graph contains either a large clique or a large independent set. Here similar dichotomic phenomena are investigated in the context of finite metric spaces. Namely, statements are provided of the form ‘every finite metric space contains a large subspace that is nearly equilateral or far from being equil...

The ProtoNet site provides an automatic hierarchical clustering of the SWISS-PROT protein database. The clustering is based on an all-against-all BLAST similarity search. The similarities' E-score is used to perform a continuous bottom-up clustering process by applying alternative rules for merging clusters. The outcome of this clustering process i...

In this note, we consider the metric Ramsey problem for the normed spaces # p . Namely, given some 1 1, and an integer n, we ask for the largest m such that every n-point metric space contains an m-point subspace which embeds into # p with distortion #. In [1] it is shown that in the case of # 2 , the dependence of m on # undergoes a phase transiti...

Abstract. Bourgain [1] showed that every embedding of the complete binary tree of depth n into l 2 has metric distortion ≥ . An alternative proof was later given by Matousek [3]. This note contains a short proof for this fact.

Motivation: A large fraction of biological research concentrates on individual proteins and on small families of proteins. One of the current major challenges in bioinformatics is to extend our knowledge to very large sets of proteins. Several major projects have tackled this problem. Such undertakings usually start with a process that clusters all...

. Let G=(I n ,E) be the graph of the n-dimensional cube. Namely, I n ={0,1} n and [x,y]∈E whenever ||x−y||1=1. For A⊆I n and x∈A define h A (x) =#{y∈I n A|[x,y]∈E}, i.e., the number of vertices adjacent to x outside of A. Talagrand, following Margulis, proves that for every set A⊆I n of size 2 n−1 we have for a universal constant K independe...

Let V be an rn-dimensional linear subspace of Z2n. Suppose the smallest Hamming weight of non-zero vectors in V is d. (In coding-theoretic terminology, V is a linear code of length n, rate r and distance d.) We settle two extremal problems on such spaces. First, we prove a (weak form) of a conjecture by Kalai and Linial and show that the fraction o...

Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information. Metric spaces also come up in many recent advances in the theory of algorithms. Finally, finite submetrics of c...