numerativeombinatorics

pplications

A Enumerative Combinatorics and Applicationsecajournal.haifa.ac.il

ECA 1:1 (2021) Interview #S3I2

Interview with Noga AlonToufik Mansour

Noga Alon graduated from the Hebrew Reali School in 1974,and received his Ph.D. in Mathematics at the Hebrew Univer-sity of Jerusalem in 1983. After his Ph.D., he spent the follow-ing two years in MIT and then joined Tel Aviv University in1985. He had visiting positions in various research institutesincluding MIT, Harvard, the Institute for Advanced Study inPrinceton, IBM Almaden Research Center, Bell Labs, Bellcoreand Microsoft Research. He is currently a Professor of Math-ematics at Princeton University and a Baumritter ProfessorEmeritus of Mathematics and Computer Science at Tel AvivUniversity, Israel. He has made significant contributions tocombinatorics and theoretical computer science. He has givenlectures in many conferences, including plenary addresses inthe 1996 European Congress of Mathematics and in the 2002

International Congress of Mathematicians. Alon has received a number of awards, includingthe Erdős Prize, the Feher Prize, the Pólya Prize, the Bruno Memorial Award, the LandauPrize, the Gödel Prize, the Israel Prize in Mathematics, the EMET Prize in Mathematics, theDijkstra Prize, the Nerode Prize and the Kanellakis Award. He has been a member of the IsraelAcademy of Sciences and Humanities since 1997, and is also a member of Academia Europaeaand of the Hungarian Academy of Sciences. He is an AMS Fellow, and an ACM Fellow, andserves as the editor-in-chief of Random Structures and Algorithms.

Mansour: Professor Alon, first of all we wouldlike to thank you for accepting this interview.Would you tell us broadly what combinatoricsis?

Alon: Combinatorics is primarily the mathe-matics of finite objects, investigating the prop-erties of combinatorial structures. Its areas ofstudy include exact and asymptotic enumera-tion, graph theory, probabilistic and extremalcombinatorics, designs and finite geometries.The subject has applications in other areas in-cluding geometry, number theory, optimizationand theoretical computer science.

Mansour: What do you think about the de-velopment of the relations between combina-torics and the rest of mathematics?

Alon: The relations between combinatorics

and other mathematical and scientific areashave been crucial in the development of themodern theory. Combinatorial concepts andquestions appear naturally in many branches ofmathematics, and the area has found applica-tions in other disciplines as well. These includeapplications in information theory and electri-cal engineering, in statistical physics, in chem-istry and molecular biology, and, of course, incomputer science. Combinatorial topics suchas Ramsey theory, combinatorial set theory,matroid theory, extremal graph theory, combi-natorial geometry and discrepancy theory arerelated to a large part of the mathematical andscientific world, and these topics have foundnumerous applications in other fields.

Mansour: Why is there a strong relationship

The authors: Released under the CC BY-ND license (International 4.0), Published: October 19, 2020Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is tmansour@univ.haifa.ac.il.

Interview with Noga Alon

between combinatorics and theoretical com-puter science?

Alon: The core of most of the fundamen-tal questions in theoretical computer scienceis combinatorial, as the notion of computationis based on manipulations with finite struc-tures. The investigation of the limits of compu-tation leads to basic combinatorial questions,and much of the design and analysis of efficientalgorithms is also combinatorial in nature.

Mansour: What have been some of the maingoals of your research?

Alon: In my research I tried (and still try) todevelop effective techniques by tackling inter-esting problems. Solving such problems, espe-cially ones with a history, is an important goal,but an even more important goal is the intro-duction of novel ideas and tools that can leadto further progress.

Mansour: What were your early experienceswith mathematics? Were there specific prob-lems that made you first interested in combi-natorics?

Alon: From an early age I have been inter-ested in solving mathematical puzzles; I par-ticipated in several mathematics competitionsfor high school students in Israel and read pop-ular mathematical books. An early encounterwith a mathematical proof that impressed mewas when I was in high school and first learnedabout the probabilistic method. I read a ver-sion of one of the earliest results establishedusing it: the proof of the lower bound for Ram-sey numbers discovered in 1947 by Paul Erdős,the founder of the method. I still rememberthe admiration I felt going through the conciseand elegant argument, an admiration that onlyincreased when I kept following the profoundimpact of other applications of the method onthe development of discrete mathematics.

Mansour: We would like to ask you aboutyour formative years. What was the first rea-son you become interested in mathematics andspecifically combinatorics? Did that happenunder the influence of your family, or someother people?

Alon: As already mentioned, I have been in-terested in mathematical puzzles since I was achild. My parents always encouraged me andmy brother to pursue whatever we found ap-pealing, with no special emphasis on exact sci-ences. During my final two years in high school

I had a superb teacher of mathematics, a newimmigrant from Ukraine, who taught us ad-vanced material after the usual hourly classes.In my final year in high school, I also met PaulErdős who has been visiting Israel frequently,and heard several questions from him that Ifound fascinating.

Mansour: What was the reason you chosethe Hebrew University of Jerusalem for yourPh.D., and your advisor, Micha Asher Perles?

Alon: I finished my undergraduate degreeat the Technion, M.Sc. at Tel Aviv Univer-sity, and Ph.D. at the Hebrew University ofJerusalem, hence, I had some experience as astudent in all three institutions. However, Icompleted both the M.Sc. and Ph.D. degreesduring my army service in the academic re-serve, when I have only been able to cometo the university once a week. I first heardabout Micha Perles from Meir Katchalski, whotaught me at the Technion, and after meetinghim, when I was about to finish my undergrad-uate degree, I became convinced that I want todo my research work with him. Micha has infact been my M.Sc. and my Ph.D. advisor, andI learned a lot from him, his profound inter-est in the close connection between combina-torics and geometry, and his general approachto mathematical research.

Mansour: What was the problem you workedon in your thesis?

Alon: In my M.Sc. thesis, I worked on aproblem that originated from a discussion Ihad with Erdős, who suggested a special case.The problem was to determine or estimate themaximum possible number of copies of a fixedgraph H in a (simple) graph with a given num-ber of edges. It turned out that the parameterdetermining the asymptotic behavior of thismaximum is the fractional cover (or match-ing) number of the graph. When I workedon the problem, I was not aware of much ofthe existing results about this notion, whichI redefined, and although I could have savedquite some time had I been more familiar withthe known literature, this ignorance did leadto some interesting new discoveries. The re-sults obtained appear in my very first paper.My Ph.D. thesis includes a chapter with anextension of this problem and also deals withadditional extremal combinatorial problems.

Mansour: How was the mathematics in Is-

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Interview with Noga Alon

rael and specifically at the Hebrew Universityof Jerusalem at the time?Alon: Israel has traditionally always beena strong center of mathematical research.The Hebrew University has been particularlystrong in Set Theory, Logic and ErgodicTheory. As mentioned, I used to come tothe university only once a week. We hadan active combinatorics seminar with othergraduate students and young researchers whoare roughly of my generation, including NatiLinial, Gil Kalai, Yaakov Kupitz, Roy Meshu-lam and Ron Adin. I still have fond memoriesfrom this period.Mansour: Would you discuss for a little bitabout your most influential results and whythey have been influential?Alon: The work on expander graphs and theirapplications, focusing on the tight relation be-tween the spectral properties of a graph and itsstructural properties, has been influential dueto the numerous applications of expanders intheoretical computer science and coding the-ory and the more recent interest in their studyin group theory and number theory.

The investigation of derandomization tech-niques, that is, techniques for converting ran-domized algorithms into deterministic ones, in-cluding the Color Coding method with Yusterand Zwick that became an important tool inwhat’s called Parameterized Complexity.

The work with Matias and Szegedy is thefoundational paper on streaming algorithmswhich played a major role in creating this ac-tive area.

The Combinatorial Nullstellensatz is a pow-erful algebraic technique with applications inGraph Theory, Combinatorics and AdditiveNumber Theory.

The work on the regularity lemma of Sze-merédi, its extensions and applications in prop-erty testing, stimulated research on the struc-ture of large graphs and is closely related to theinvestigation of convergent graph sequences.

The book with Spencer is the leading texton probabilistic methods in combinatorics,which is a very active area in modern com-binatorics.

Besides these, there are several papers thatsettle long standing problems including a prob-lem of Shannon on the zero-error capacity ofthe disjoint union of two channels, and the

Hadwiger-Debrunner (p, q)-problem in Combi-natorial Geometry.Mansour: What would guide you in your re-search? A general theoretical question or aspecific problem?Alon: I usually like to think about specificproblems in the hope that progress in theirstudy will lead to the development of novelmethods and provide new insights on moregeneral topics.Mansour: When you are working on a prob-lem, do you feel that something is true evenbefore you have the proof?Alon: This varies. Sometimes I am sure Iknow the correct answer even before I see aproof, and sometimes I don’t know whether astatement is true or not and I spend compa-rable amount of time trying to find a proofand trying to think about counter-examples.During the years, however, it happened to meseveral times that I was convinced about thecorrect answer before I knew a proof and thisturned out to be wrong. I sometimes foundsuch unexpected counterexamples by myself,while trying to prove the converse, and some-times learned about such examples by otherresearchers. I am thus trying to always remindmyself that even a strong intuition about thevalidity of a statement before there is an ac-tual proof should always be taken with a grainof salt, taking into account the possibility thatthe intuition may be misleading.Mansour: What three results do you considerthe most influential in combinatorics duringthe last thirty years?Alon: This is not an easy question, and I cancome up with several different choices. Here isone possibility:

The Graph Minors monumental project ofRobertson and Seymour.

The Interlacing Polynomials method ofMarcus, Spielman and Srivastava which led tothe proof of existence of bipartite Ramanujangraphs of all degrees, and to the solution of theKadison-Singer problem and its variants.

The proof of Keevash that the obvious nec-essary divisibility conditions for the existenceof designs are also sufficient, for all values ofthe parameters, provided the number of ele-ments of the ground set is sufficiently large,settling asymptotically the main problem indesign theory raised by Steiner in the 19th cen-

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Interview with Noga Alon

tury.Mansour: What are the top three open ques-tions in your list?Alon: Here, too, I can give several answers,here is one:

The P versus NP problem, formulated asa question in Circuit Complexity (whose solu-tion could be even stronger than solving the Pversus NP problem).

Finding a proof of the Four Color Theoremwhich is not computer-aided, possibly by usingalgebraic tools.

Deciding if the maximum possible Shannoncapacity of a graph with independence number2 is bounded. Equivalently: deciding if there isa finite constant C so that the maximum pos-sible number of vertices in a complete graphwhose edges can be colored by k colors withno monochromatic triangle is at most Ck (thisis sometimes called the Schur-Erdős problem).Mansour: What kind of mathematics wouldyou like to see in the next ten-to-twenty yearsas the continuation of your work?Alon: I would love to see new applicationsand further development of my Combinato-rial Nullstellensatz, which has already beenextended in various directions. It has beenused extensively in Combinatorics, Graph the-ory, and Additive Number Theory, but I be-lieve that additional development may lead tomore sophisticated applications. Progress hereshould probably combine combinatorial rea-soning with tools from algebraic geometry.

I have already seen fascinating extensionsof some of my other results, like the necklacesplitting theorem, the (p, q)-theorem in combi-natorial geometry and the research of stream-ing algorithms, and it is always rewarding tosee more. I also hope, of course, to play anactive role in at least some of these potentialfuture developments.Mansour: What would you say about some ofthe major directions in combinatorics for thenext two decades?Alon: The tight connection with theoreticalcomputer science suggests that an algorithmicpoint of view, and maybe also computer aidedproofs, will play a major role in the future de-velopments of the subject. I also believe thatmodern combinatorics will use even more toolsfrom other branches of mathematics in the fu-ture. Applications of combinatorial techniques

in Number Theory and Statistical Physics arevery active already and are likely to keep de-veloping. I also hope to see directions that arehard to predict at the moment.

Mansour: Do you think that there are core ormainstream areas in mathematics? Are sometopics more important than others?

Alon: There are always areas in mathemat-ics that are considered mainstream, but thesekeep changing. Moreover, some of the most ex-citing mathematical results combine differentareas, and are based on sometimes unexpectedrelations between seemingly unrelated fields. Ithus believe that mathematics should be con-sidered as one unit, and we should not try toidentify the important and less important ar-eas.

Mansour: What do you think about the dis-tinction between pure and applied mathemat-ics that some people focus on? Is it meaningfulat all in your own case? How do you see therelationship between so-called “pure” and “ap-plied” mathematics?

Alon: In his book, “A Mathematician’s Apol-ogy”, Hardy stresses his opinion that the mostbeautiful mathematics is pure mathematics,that has no practical applications in the out-side world. He gives his own special field, num-ber theory, as an example of a topic that wouldnever have any practical applications. As weknow now, Number Theory, prime numbersand algorithms for factorization, play a majorrole in Cryptography, which is one of the mostapplicable mathematical areas. This indicatesthat the border between “pure” and “applied”mathematics is very unclear, and keeps chang-ing. Personally, I like to work on interestingmathematical problems without thinking muchabout their potential practical applications, ifany. But if some of the results do lead to prac-tical applications (as happened to me with thecase of streaming algorithms), I view it as anextra merit.

Mansour: What advice would you give toyoung people thinking about pursuing a re-search career in mathematics?

Alon: Think about the problems and the areasyou find interesting, without trying to identifythe “important” ones. The key to success is tobe enthusiastic about your subject. In addi-tion, try to learn as much mathematics as youcan early. It is much harder to learn new topics

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Interview with Noga Alon

later, and having a strong general mathemati-cal background is indispensable.

Mansour: Would you tell us about your in-terests besides mathematics?

Alon: Reading, traveling around the world,playing table tennis (enthusiastically even ifnot at a very high level).

Mansour: Before we close this interview withone of the foremost experts in combinatorics,we would like to ask some more specific math-ematical questions:

You have worked on many problems fromcombinatorics and theoretical computer sci-ence throughout your career. Which one isyour favourite?

Alon: The application of basic properties ofpolynomials in these areas. This includes theconstruction of small sample spaces supportingk-wise independent random variables, which isuseful in the derandomization of randomizedalgorithms. It also includes a convenient way Ifound for bounding the Shannon capacity of agraph that led to a solution of a problem raisedby Shannon in 1956. Notably, it also includesthe Combinatorial Nullstellensatz, a variant ofHilbert’s fundamental Nullstellensatz which isthe basis of Algebraic Geometry. The Combi-natorial Nullstellensatz found applications inCombinatorics, Graph Theory and AdditiveNumber Theory. When it works it often givesoptimal results and usually hardly requires anytedious computation.

Mansour: Your book with Joel Spencer is themain reference for the probabilistic method.What is the probabilistic method? Is therea result that you consider the most interest-ing obtained by the probabilistic method? Anycomments on the research in this direction?

Alon: The probabilistic method is a powerfultechnique for proving deterministic statementsby showing that a random structure or sub-structure of a given one in an appropriatelydefined sample space satisfies a set of desiredproperties with positive probability, and there-fore exists. Although this may sound naive, thepower of the technique is that it enables one toapply techniques from probability theory in thestudy of problems in combinatorics, numbertheory, geometry and more. One of the first ap-plications of the probabilistic method in com-binatorics is the proof of Erdős, the founder ofthe method, of an exponential lower bound for

the diagonal Ramsey numbers: for every inte-ger t > 2 there are graphs with at least 2t/2

vertices that contain neither a clique nor anindependent set of size t. Although, by now,this short and simple proof appears in the firstchapter in any text-book on the subject, itssubtlety can be appreciated by the fact thatwe still do not know any efficient algorithmfor constructing such graphs. A more sophis-ticated result obtained by the method is theexistence of graphs with high girth and highchromatic number. The method is one of themost powerful and widely used tools in modernCombinatorics and its applications. I believeand hope that our book contributed to the suc-cess and popularity of the subject.

Mansour: How would you describe the ex-tremal combinatorics? What would you sug-gest for a graduate student and young re-searcher who wants to work in this field?

Alon: Extremal Combinatorics, in its strictestsense, deals with the problem of determiningor estimating the maximum or minimum pos-sible cardinality of a collection of finite ob-jects that satisfies certain requirements. Thearea is, however, much broader, including in itsscope the investigation of inequalities betweencombinatorial invariants, and questions deal-ing with relations among them. Extremal com-binatorial problems are often related to otherareas including Computer Science, InformationTheory, Number Theory and Geometry. Thisbranch of Combinatorics has developed spec-tacularly over the last few decades. A youngresearcher interested in the field can start byreading some of the books or surveys on thesubject. The book of Jukna, extremal com-binatorics with applications in computer sci-ence, with its many references is one excellentsource.

Mansour: What is an expander? Why didthey become so popular? Any comments onthe research in this direction?

Alon: Expanders are sparse graphs withstrong connectivity properties. A formal def-inition for a family of expanders is an infi-nite family of d-regular graphs, where for eachmember of the family, every set U of at mosthalf of its neighbors has at least c|U | neigh-bors outside the set. The crucial fact here isthat d and c > 0 are fixed for all the graphsin the family. Such graphs with various val-

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Interview with Noga Alon

ues of the parameters have numerous applica-tions, some, like the construction of intercon-nection networks, are natural given the defi-nition, but some are much less expected andinclude derandomization techniques, construc-tion of error correcting codes, the design of ef-ficient sorting networks, and more. The rea-sons for the popularity of the subject are notonly the applications but also the fact that thestudy of the subject led to the introduction ofmany interesting techniques and to the discov-ery of tight connections it has to deep resultsin Number Theory and Group Theory. Thearea is still very active, with exciting ques-tions about explicit constructions, about thecombinatorial pseudo-random properties of ex-panders and about higher dimensional exten-sions.

Mansour: In your 1987-paper on “Splittingnecklaces”∗, you gave a short beautiful proofby using topological results. What do youthink about the importance of being compe-tent in different branches of mathematics?

Alon: Many of the most exciting results inMathematics are obtained by combining toolsfrom seemingly unrelated areas. It is, there-fore, crucial to be as knowledgeable as possi-ble in diverse mathematical areas, even if oneis interested mainly in applications in a morerestricted subject. In my own research, I man-aged several times to use tools and intuitionfrom other areas in the study of combinato-rial problems. Examples include the applica-tion of Grothendieck’s Inequality in the devel-opment of efficient algorithms for large graphs,using character sum estimates in the study ofCayley graphs, relying on the intuition arisingfrom classical isoperimetric inequalities in thestudy of expanders, applying results in real al-gebraic geometry including theorems of Milnorand Warren in the investigation of sign rankof matrices and related problems, and the ap-plication of topological, algebraic, probabilisticand spectral techniques in discrete mathemat-ics and in theoretical Computer Science. Thereare, of course, lots of important mathematicalareas in which my knowledge is very limited.One reminder I got of this fact was when Iserved as the chair of the scientific committeeof the 2006 international congress of mathe-maticians. I wish I had a better global knowl-

edge of mathematics, I am trying regularly tohear at least some lectures focused on areas inwhich I am far from being an expert, and be-lieve that it is desirable to learn the basics ofdifferent branches of mathematics already atthe beginning of a mathematical career.

Mansour: Mathematicians have been work-ing on graphs for centuries and have devel-oped several tools to understand their struc-tures. Suppose someone comes with a “large”graph and asks (naively) your help to under-stand the structure of her graph. How wouldyou help her? What are the basic tools she canuse?

Alon: The regularity lemma of Szemerédi andits variants form a basic tool in understand-ing the structure of large (dense) graphs. Al-though the dependence of the statements onthe parameters is highly non-practical, the in-sight the lemma and its applications in what’sknown as graph property testing provide is use-ful. Property testing has been investigatedintensively for sparse graphs as well, showingthat sampling techniques can often provide alot of information about a large graph, evenwhen we sample only a tiny fraction of it.

Mansour: What kind of enumerative prob-lems look interesting to you? Why is it verydifficult to count some combinatorial objects?Do you think that mathematicians need to in-vent some new tools to count such objects orjust use the existing tools in an ingenious way?

Alon: It is a fascinating fact that there areproblems in enumerative combinatorics thatare very simple to state and look extremelyhard to solve. Recall that for n > k a permu-tation π ∈ Sn contains a permutation σ ∈ Skif the permutation matrix of π contains thatof σ as a submatrix. If π does not containσ, it is called σ-avoiding. What is the num-ber of 1324-avoiding permutations? Does ithave a nice recurrence relation, and why is thisproblem so much more difficult than the wellunderstood case of avoiding a permutation oflength 3? What is the number of 132-avoidingcyclic permutations? Is it roughly the totalnumber of 132-avoiding permutations dividedby n? Is the number of integers n between 1and N for which the partition function p(n)is even (1/2 + o(1))N? To me, personally, theasymptotic problems are often more interesting

∗https://www-sciencedirect-com.ezproxy.haifa.ac.il/science/article/pii/0001870887900557?via%3Dihub

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Interview with Noga Alon

than the ones dealing with precise counting. Aspecific (admittedly somewhat special) prob-lem I found (and still find) interesting, moti-vated by a question about Hamilton cycles intournaments, is the asymptotic behavior of thenumber of cyclic permutations (a0, a1, . . . , a2k)of Z2k+1 in which for every i the difference(ai+1 − ai) modulo 2k + 1 is at most k. Is thisnumber much larger than Qk = (2k)!/2

2k+1 forlarge k? That is, does the ratio between thisnumber and the quantity Qk tend to infinityas k tends to infinity?

Mansour: Would you tell us about yourthought process for the proof of one of yourmost important results? How did you becomeinterested in that problem? How long did ittake you to figure out a proof? Did you havea “eureka moment”?

Alon: There are results that are among mybest on which I spent a very long time. Ihave worked on expanders for years, and infact even recently investigated the delocaliza-tion of eigenvectors of high girth expanders.This involved reading a lot, and thinking alot over the years, discussing the subject withother researchers and returning to it time andagain. Similarly, I spent a long time workingon different variants of the polynomial methodthat yielded the combinatorial nullstellensatz(although the basic end result is not compli-cated). Here, the main results evolved with theinvestigation of a considerable number of par-ticular problems. There are, however, resultsthat I believe are important in which there hasbeen a very clear and quick “eureka moment”.Our paper with Kleitman in which we solvedthe Hadwiger-Debrunner (p, q)-problem raisedin the 50s was a result obtained in essentiallyone day. Kleitman was visiting me in Tel Aviv,we heard a seminar lecture in which someonementioned the problem (that I did hear be-fore), and we had the main idea in that after-noon. Finishing the proof took a bit longer,but it was clear already in that day that weknow how to do it. I had a similar experi-ence with a paper with Rödl settling a Ram-sey theoretic problem raised by Erdős and Sósin 1979. This was during an Oberwolfach con-ference in 2002, someone mentioned this prob-lem during the open problems session, and wecame with the key idea almost immediately.Here, too, this was a problem we have both

heard before. Naturally, the detailed proof re-quired more work that took a few weeks, butthe basic crucial idea was nearly instantaneous.It sometimes happens that you see the rightapproach for tackling a problem immediately,possibly because you have seen recently some-thing relevant, or thought about a closely re-lated topic, and it may well be the case that inanother day and another state of mind you willhave no clue considering the very same ques-tion. My final example of a quick solution isa paper I have written around 1990, solvinga problem of Erdős and Purdy. My colleagueAmos Fiat, who is now a well known computerscientist, met me in the corridor of the math-ematics building in Tel Aviv University andshowed me a book of Richard Guy: UnsolvedProblems in Number Theory, Springer, 1981,that he received from his advisor Adi Shamir.I was on my way to a colloquium lecture, soI just took a quick glance at the book, open-ing it at a random place and reading a prob-lem that looked interesting and easy to state.During the lecture, I figured out an approachthat seemed promising for tackling the prob-lem, and later that day, I basically had the fullsolution. I went back to Amos, showed himthe solution and asked to take a closer look atGuy’s book, as it seemed likely, based on thesingle sample I observed, that I should be ableto solve a large fraction of the problems in thisbook. I then spent a week going carefully overthe problems in the book, one by one, and hadno idea how to make any progress in the inves-tigation of any of them, besides the first one Ihave initially picked. I don’t claim that thereis any meaningful moral here, but think it isan interesting (and accurate) story.

Mansour: Is there a specific problem youhave been working on for many years? Whatprogress have you made?

Alon: I have been working on questions thatdeal with expander graphs for many years.It is not difficult to show that random regu-lar graphs are typically very good expanders,much better than the best known expanderswe know to construct explicitly. The explicitconstructions use the connection between theexpansion properties of a regular graph andits second largest eigenvalue, a connection thatappears in my work with Milman and in otherpapers. This approach cannot lead to ex-

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Interview with Noga Alon

panders that are nearly as good as the ran-dom ones, as the spectral approach is lim-ited by a lower bound on the second largesteigenvalue of any large bounded degree regulargraph, that I established with Boppana. I (andmany others) have been thinking on and off foryears about alternative ways to prove expan-sion, but, at the moment, there is no known

approach that gives explicit constructions thatare as good as the random ones. Such con-structions will have interesting applications intheoretical computer science and other areas.Mansour: Professor Noga Alon, I would liketo thank you for this very interesting interviewon behalf of the journal Enumerative Combi-natorics and Applications.

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