the mathematics of paul erdos · 1999-03-08 · paul erdo˝s was also a kind and generous man, one...

JANUARY 1998 NOTICES OF THE AMS 19

The Mathematics ofPaul Erdos

Laszló Babai, Carl Pomerance, and Péter Vértesi

Paul Erdos died September 20, 1996, anda memorial article appears elsewhere inthis issue. This feature article gives a crosssection of his monumental oeuvre. Mostof Erdos’s work falls roughly into the fol-

lowing categories:• number theory• finite combinatorics (including graph theory)• combinatorial geometry• set theory, set-theoretical topology• constructive theory of functions (approxima-

tion theory)• other areas of classical analysis (polynomials,

theory of series, functions of a complex vari-able)

• probability theory, ergodic theoryThe first two areas are represented in Erdos’s

work by more than 600 articles each, the nextthree by more than 100 articles each. There aresome overlaps in this rough count. A large num-ber of articles fall into a “miscellaneous” category.

In what follows, Pomerance gives a glimpse intothe variety of topics Erdos worked on in numbertheory. Babai discusses (infinite) set theory, finitecombinatorics, combinatorial geometry, combina-torial number theory, and probability theory.Vértesi treats approximation theory, with a hint ofrelated work on polynomials.

Paul Erdos, NumberTheoristExtraordinaireCarl Pomerance

Nearly half of Paul Erdos’s 1,500 papers were innumber theory. He was a giant of this century,showing the power of elementary and combinato-rial methods in analytic number theory, pioneer-ing the field of probabilistic number theory, mak-ing key advances in diophantine approximationandarithmetic functions, and until his death leadingthe field of combinatorial number theory. PaulErdos was also a kind and generous man, one whowould seek out young mathematicians, work withthem, give them ideas, teach them, and in theprocess make a lifelong friend and colleague. Iwas one of these lucky ones, but more on thatlater.

Perhaps the single most famous paper of Erdosis [3], wherein he described an elementary proofof the prime number theorem.

The history of the prime number theorem seemsto be punctuated by major developments at half-century intervals and often in two’s. At the end ofthe eighteenth century Gauss and Legendre inde-pendently conjectured that the number of primesup to x, denoted π (x), is asymptotically x/ logxas x→∞ . (This came some fifty years after Eulerhad proved that the sum of the reciprocals of theprimes is infinite.) In the mid-nineteenth centuryChebyshev showed by an elementary method thatthere are positive constants c1 , c2 withc1x/ logx < π (x) < c2x/ logx for all large x, and

—László Babai, Organizer

Carl Pomerance is research professor of mathematics atthe University of Georgia, Athens. His e-mail address [email protected].

Note: Except where otherwise noted, allphotographs in this article are from the col-lection of Vera T. Sos.

20 NOTICES OF THE AMS VOLUME 45, NUMBER 1

Riemann laid down a plan to prove the prime num-ber conjecture of Gauss and Legendre via analyticmethods. It is in this paper that Riemann statedwhat came to be known as the “Riemann hypoth-esis,” one of the most famous and important un-solved problems in mathematics.

At the close of the nineteenth century, de la Val-lée Poussin and Hadamard independently suc-ceeded in giving complete proofs of the primenumber theorem. (Though roughly following Rie-mann’s plan, they avoided an outright assault onthe Riemann hypothesis.) A tour de force for ana-lytic methods in number theory, it was thought bymany that an elementary proof of the prime num-ber theorem was impossible. It was thus quite a sen-sation when Erdos and Atle Selberg actually didcome up with elementary proofs in 1948.

Another fifty years have passed. Will there soonbe another great advance?

In response to the theory of quantum mechan-ics, Einstein exclaimed, “God does not play dicewith the universe.” Though this never happened,I would like to think that Paul Erdos and the greatprobabilist, Mark Kac, replied, “Maybe so, but some-thing is going on with the primes.” In 1939 Erdosand Kac [10] proved one of the most beautiful andunexpected results in mathematics. Their theo-rem states that the number of prime factors in anumber is distributed, as the number varies, ac-cording to a Gaussian distribution, a bell curve.

Let ω(n) denote the number of distinct primefactors of n. In 1917 Hardy and Ramanujan provedthat ω(n) is normally log logn. What this meansis that for each ε > 0, the density of the set of nat-

ural numbers n with (1− ε) log log n < ω(n) <(1 + ε) log logn is 1. (A set S of natural numbershas density d if the number of members of S upto x, when divided by x, tends to d as x→∞ . So,for example, the odd numbers have density 1/2,the prime numbers have density 0, and the set ofnumbers with an even number of decimal digitsdoes not have a density.) Later, Paul Turán, a closefriend of Erdos, came up with a greatly simplifiedproof of the Hardy-Ramanujan theorem by show-ing that the sum of (ω(n)− log logn)2 for n up tox is of order of magnitude x log logx . This wouldlater come to be thought of as a variance calcula-tion as in probability theory, but it was not con-ceived of in this way.

Mark Kac viewed the number theoretic func-tion ω(n) probabilistically. He reasoned that beingdivisible by 2,3,5, etc., should be thought of as “in-dependent events,” and so ω(n) could be viewedas a sum of independent random variables. Sincethe sum of 1/p for p prime, p ≤ x , is aboutlog logx, Kac reasoned that this is what is behindthe Hardy-Ramanujan-Turán theorem and that infact a Gaussian distribution should be involved,with standard deviation

√log logx. That is, Kac

conjectured that for each real number u, the den-sity of the set of n with ω(n) ≤ log logn+u√

log logn exists and is equal to

1√2π

∫ u−∞e−t2/2dt,

the area under the bell curve from −∞ to u.

How the collaboration of Erdos and Kac cameabout is best left to Kac’s own words, as quotedby Peter Elliott in [2]:

“If I remember correctly, I first stated (as a conjec-ture) the theorem on the normal distribution of thenumber of prime divisors during a lecture in Prince-ton in March 1939. Fortunately for me and possi-bly for Mathematics, Erdos was in the audience, andhe immediately perked up. Before the lecture wasover he had completed the proof, which I could nothave done, not having been versed in the numbertheoretic methods, especially those related to thesieve.”

What Erdos knew quite well was that via themethods of sieves, as developed by Brun early inthis century, it could be shown that the primes upto xε actually do distribute themselves “indepen-dently” among the numbers up to x, and of courseno number n ≤ x can have more than 1/ε primefactors that exceed xε.

This result opened the book on probabilisticnumber theory, the branch of mathematics thatstudies number theoretic functions, such as ω(n),via probabilistic methods.

The Manchester bridge party (1934–1938).Bottom to top: Harold Davenport, Erdos,Chao-Ko, Zilinkas.

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Paul Erdos is equally well known for his re-markable problems. Here are a few in number the-ory:

1. Among the integers up to x, the set of powersof 2 have the property that the various subsetsums are all different, and of course there arelog2 x +O(1) powers of 2 up to x. Is it true that ifS is a set of integers in [1, x] with all subset sumsdifferent, then S has at most log2 x +O(1) mem-bers?

2. Suppose A is a subset of the nonnegative inte-gers such that every nonnegative integer n can bewritten as a1 + a2, where a1, a2 are in A . Let r (n)be the number of such representations of n. Mustthe sequence r (n) be unbounded?

3. Suppose B is a subset of the positive integerswith the sum of the reciprocals of the members ofB being infinite. Must B contain arbitrarily longarithmetic progressions?

4. The set {2,3,4,6,12} is a “covering set”, sincethere are residue classes with these moduli thatcover the integers: in particular, 0 mod 2, 0 mod 3,1 mod 4, 1 mod 6, 11 mod 12 will do. For each kis there a (finite) covering set with distinct mod-uli, each > k?

Each of these problems has its own interestingstory, as do hundreds of other Erdos problems.They are often tips of icebergs. For example, prob-lem 2, which is joint with Turán, is related to thefamous theorem of Erdos and W. Fuchs [8], whichasserts that no matter what sequence is chosen forA , the sum of r (n) for n up to x cannot be of theform cx + o(x1/4/(logx)1/2). (Erdos was justifiablyproud of this beautiful result. A few years ago hewrote (in [7]) that the Erdos-Fuchs theorem “cer-tainly will survive the authors by centuries.”)

Problem 4 is related to an old problem of Euler,who considered whether an odd number n > 1can be expressed as a sum of a prime and a powerof 2. Euler noticed that 127 and 959 cannot be rep-resented, though de Polignac conjectured in 1849that every odd number n > 1 can be represented!The problem was revived by Romanoff in 1934 andsolved independently by Erdos [4] and van derCorput [1] in 1950. In fact, Erdos showed thatthere is an infinite arithmetic progression of oddnumbers that cannot be represented as a sum ofa prime and a power of 2. The proof used the cov-ering set {2,3,4,8,12,24}. If problem 4 were tohold, then there would be, for each positive inte-ger k, an infinite arithmetic progression contain-ing no numbers that are a sum of a power of 2 anda number with at most k different prime factors.Erdos was fond of repeating Selfridge’s coveringset problem: is there a covering set with odd mod-uli > 1? This is still unsolved.

While he was alive, Erdos offered money foreach of problems 1–4 and many others. For ex-ample, problem 3 (which would have the sensa-tional corollary that there are arbitrarily long arith-metic progressions consisting of primes) was worth$3,000. Erdos liked to joke that his prize moneyviolated the minimum wage law.

Paul Erdos, often through his prizes, inspiredmany other mathematicians. Endre Szemerédi, forexample, earned $1,000 when he showed a slightlyweaker result than problem 3: he showed that ifB does not have density 0, then it contains arbi-trarily long arithmetic progressions. And HelmutMaier and Gérald Tenenbaum earned money fromErdos when they showed that the density of thosenumbers nwith two divisors a, b with a < b < 2ais 1.

I too owe much to Paul Erdos. At the end of anarticle in 1956 Erdos gave a brief heuristic argu-ment on why he thought there should be infinitelymany Carmichael numbers and, in fact, why theyshould be plentiful among all numbers. (A com-posite number n for which an ≡ a mod n for alla is called a Carmichael number . In 1910Carmichael conjectured there should be infinitelymany.) Erdos and I discussed his heuristic argu-ment several times over the years, and he was very

Erdos in deep thought, around 1960.


pleased when Red Alford, Andrew Granville, andI succeeded recently in making it the backbone ofa proof of the infinitude of Carmichael numbers.We were happy to dedicate our paper to Erdos onthe occasion of his eightieth birthday.

I like to tell the story of how I first met PaulErdos, since it is not only a good story, but showsa fundamental quality of Erdos as a man and amathematician. I was home on April 8, 1974, watch-ing a baseball game on television. I was then an as-sistant professor at the University of Georgia, lessthan a couple of years from graduate school, withfew theorems but a love of numbers. This was notan ordinary baseball game, but the one in whichHank Aaron of the Atlanta Braves hit his 715thmajor league home run, thus surpassing the sup-posedly unbeatable record of 714 that had beenset by Babe Ruth some four decades earlier.

I noticed that 714 and 715 have a peculiar prop-erty, namely, that their product is also the prod-uct of the first 7 primes. The next morning I chal-lenged my colleague David Penney to find aninteresting property of 714 and 715. He soon foundthe same thing I had, but he also posed the prob-lem to his numerical analysis class, where a stu-dent came up with another interesting property:the sum of the prime factors of 714 is equal to thesum of the prime factors of 715. Working with an-other student, Carol Nelson, Penney and I foundmany other examples of consecutive pairs of num-bers with this latter property and were able tocome up with a strong heuristic for why thereought to be infinitely many. We wrote up our ob-servations in a light-hearted article that was pub-lished several months later in the Journal of Recre-ational Mathematics. Calling 714 and 715 a“Ruth-Aaron pair”, we conjectured that such pairshave density 0: that is, the set of n, such that thesum of the prime factors of n is equal to the sumof the prime factors of n + 1, has density 0.

Paul Erdos, the giant of twentieth-century num-ber theory, was also a reader of the Journal ofRecreational Mathematics. He did not know me, norshould he have, but he wrote me a letter saying hecould prove the conjecture that Ruth-Aaron pairshave density 0 and he would like to visit Georgiaand discuss it with me. Much of what I now knowin mathematics I learned from Erdos working onthis and subsequent joint papers. It is fair to saythat I owe my career to this serendipitous collab-oration.

I am very grateful to have this chance to writethese words in tribute to Paul Erdos. But I am cog-nizant of the vast amount of his work that did notget mentioned. In fact, I cannot close without giv-ing four more delightful results.

Amicable numbers have been studied sincePythagoras: a pair m,n is amicable if the sum ofthe proper divisors of m is n, and vice versa. Thefirst amicable pair with m 6= n is 220,284. Paul

Erdos [5] was the first to prove thatthe set of am-icable numbers has density 0. It is still not knownif there are infinitely many.

Can the product of consecutive integers be apower? This problem, with roots in the eighteenthcentury, was settled in the negative by Erdos andSelfridge [11] in 1975. It is still not known whetherthe Erdos-Selfridge theorem can be generalized toarithmetic progressions—the conjecture is that aproduct of four or more consecutive terms of a co-prime arithmetic progression cannot be a power.

An additive function f (n) is a real-valued func-tion defined on the natural numbers with the prop-erty that f (mn) = f (m) + f (n) whenever m and nare coprime. For example, the number-of-prime-fac-tors function ω(n), mentioned above in connectionwith the Erdos-Kac theorem, is additive, as is thefunction logn. In 1944 Erdos proved that if f (n)is additive and f (n + 1) ≥ f (n) for all large n, thenf (n) = c logn for some number c. This also holdsif one replaces the monotonicity assumption withf (n + 1)− f (n) → 0. Others, including Feller, Wirs-ing, Kátai, and Kovács, have added more to this the-ory of characterizing the logarithm as an additivefunction. For example, Wirsing [14] proved in 1970the long-standing conjecture of Erdos that if f (n)is additive and f (n + 1)− f (n) is bounded, thenthere is a number c such that f (n)− c logn isbounded.

Finally, I cannot resist describing the Erdos“multiplication table theorem.” Let M(n) be thenumber of distinct numbers in the n× n multi-plication table. For example, in the familiar 10× 10multiplication table (at least familiar to those ofus who did not grow up with calculators), there are43 distinct numbers among the 100 entries, andso M(10) = 43. Erdos asked about the behavior ofM(n)/n2 as n →∞. What do you think it is? Clearly,since the multiplication matrix is symmetric, wehave lim sup M(n)/n2 ≤ 1/2. Is 1/2 the limit?Erdos showed [6] in 1960 that M(n)/n2 → 0 asn →∞, a theorem that I find as surprising as it isdelightful. (Once one sees the proof, the surprisefactor diminishes, though not the delight. As wesaw before, most numbers up to n have aboutlog logn prime factors, and thus most products inthe table have about 2 log logn prime factors. Thisis an abnormal number of primes for a number upto n2, so there are not very many products.) Westill do not have an asymptotic formula for M(n)as n →∞, though from the work of Tenenbaum wehave some good estimates.

This last result illustrates a most importantpoint. At first glance one might think of the workof Erdos as a collection of unconnected and ad hocresults. Upon deeper inspection, especially of theproofs, one finds a glorious theory, with many in-terrelations of ideas and tools. It is this edifice of“Erdos-theory” that Paul Erdos leaves for us, andnumber theory is much the richer for it.


For more on the number theory of Paul Erdos,see [7, 9, 12, 13].

References[1] J. G. VAN DER CORPUT, On de Polignac’s conjecture

(Dutch), Simon Stevin 27 (1950), 99–105.[2] P. D. T. A. ELLIOTT, Probabilistic number theory, Vol.

2, Springer-Verlag, New York, 1980, p. 24.[3] P. ERDOS, On a new method in elementary number the-

ory which leads to an elementary proof of the primenumber theorem, Proc. Nat. Acad. Sci. U.S.A. 35(1949), 374–384.

[4] ———, On integers of the form 2k + p and some re-lated problems, Summa Brasil. Math. 2 (1950),113–123.

[5] ———, On amicable numbers, Publ. Math. Debrecen4 (1955), (1960), 108–111.

[6] ———, An asymptotic inequality in the theory of num-bers (Russian), Vestnik Leningrad Univ. Mat. Mekh.i Astr. 13, (1960), 41–49.

[7] ———, On some of my favourite theorems, in Com-binatorics, Paul Erdos is Eighty, Vol. 2, Keszthely(Hungary), 1993, Bolyai Math. Stud., Vol. 2, Budapest,1996, pp. 97–132.

[8] P. ERDOS and W. H. J. FUCHS, On a problem of additivenumber theory, J. London Math. Soc. 31 (1956),67–73.

[9] P. ERDOS and R. L. GRAHAM, Old and new problems andresults in combinatorial number theory, L’Enseigne-ment Mathématique, vol. 28, Imprimerie Kundig,Geneva, 1980.

[10] P. ERDOS and M. KAC, The Gaussian law of errors inthe theory of additive number theoretic functions,Amer. J. Math. 62 (1940), 738–742.

[11] P. ERDOS and J. L. SELFRIDGE, The product of consecu-tive integers is never a power, Illinois J. Math. 19(1975), 292–301.

[12] C. POMERANCE and A. SARKÖZY, Combinatorial numbertheory, in Handbook of Combinatorics (R. L. Graham,M. Grötschel, L. Lovász, eds.), Elsevier, Amsterdam,1995.

[13] A. SARKÖZY, Farewell, Paul, Acta Arithmetica (to ap-pear).

[14] E. WIRSING, A characterization of log n as an additivearithmetic function, Sympos. Math., Vol. IV (INDAM,Rome, 1968/69) Academic Press, London, 1970, pp.45–57.

Finite andTransfiniteCombinatoricsLászló Babai

The Combinatorial VeinA hallmark of much of Erdos’s work is his uniquecombinatorial vision, which revolutionized severalfields of mathematics. Wherever he looked, hefound elementary, yet often enormously difficult,combinatorial questions.

Nothing serves as a better illustration of thispoint than the excitement his questions broughtto the simplest concepts of Euclidean plane geom-etry: points, lines, triangles.

Consider a set of k points and t lines in theplane. What would Erdos ask about them? Manythings, but perhaps the simplest question is this:what is the maximum number f (k, t) of incidencesbetween the points and the lines? After many yearsSzemerédi and Trotter (1983) confirmed Erdos’sconjecture that the points of a square grid to-gether with a certain set of lines give the optimalorder of magnitude. The proof from The Book ap-peared in 1997 (L. Székely).

Another, even simpler, problem of Erdos asksthe maximum number g(n) of unit distances thatcan occur among n points in the plane. In 1946Erdos proved that n1+c/ log logn < g(n) < cn3/2. Theupper bound was improved by Beck, Spencer, Sze-merédi, and Trotter to n4/3. Erdos conjecturedthat the lower bound, obtained from the squaregrid, has the correct order of magnitude. The prob-lem remains wide open; the large gap between theupper and lower bounds offers a continuing chal-lenge.

Let me quote a related open problem Erdos vol-unteered for the “Math Investigations” column ofGeorge Berzsenyi in the student journal Quantum:“Let f (n) be the largest integer for which there isa set of n distinct points x1, x2, . . . , xn in the planefor which for every xi there are ≥ f (n) points xjequidistant from xi. Determine f (n) as accuratelyas possible. Is it true that f (n) = o(nε) for everyε > 0?” Erdos offered $500 for a proof and “muchless” for a counterexample. The estimatef (n) < cn2/5 follows from a result of Clarkson etal. (1990).

As these questions indicate, Erdos’s combina-torics, as long as finite sets are concerned, is aboutasymptotic orders of magnitude. Asymptotic think-ing has been common in number theory (espe-cially in the study of the distribution of primenumbers), which was Erdos’s first love. But it seemsto be without precedent in combinatorics and ingeometry. And even within number theory, Erdos’sstyle brought about a new field, combinatorialnumber theory, an area expounded in hundreds ofpapers by Erdos (57 of them joint with A. Sárközy).

Combinatorial ideas appear already in Erdos’searliest work on number theory. Erdos was greatlyinfluenced by a question he heard in 1934 fromFourier analyst Simon Sidon on sequences of in-tegers with pairwise different sums. In a paperpublished in Tomsk (Siberia) in 1938 Erdos con-siders a multiplicative version of Sidon’s problem:what is the maximum number f (n) of positive in-

László Babai is professor of computer science and math-ematics at the University of Chicago. His e-mail addressis [email protected].


tegers ai ≤ n such that all the pairwise productsaiaj are different? Erdo s proves thatf (n) = π (n) +O(n3/4) by reducing the problem tothe following lemma: If a graph on k vertices hasno 4-cycle, then it has at most O(k3/2) edges. Thisresult is a precursor of extremal graph theory, an-other field Erdos set out to create more than adecade later. In showing the near-optimality of hiserror term, Erdos relies on the near-optimality ofhis graph theory bound, demonstrated by formerfellow student Eszter (Esther) Klein using a pro-jective plane of prime order (which she remarkablyrediscovered from scratch).

Master of PatternsElementary geometry and Ramsey the-ory met in one of Erdos’s earliest pa-pers (1935), written with fellow un-dergraduate and lifelong friend, GeorgeSzekeres, then a student of chemicalengineering. They proved that suffi-ciently many points in the plane nec-essarily include k points that form aconvex k-gon. Later Erdos dubbed thequestion the “Happy Ending Problem”:proposed by Eszter Klein and Erdos, theproblem was first solved by Szekeres,who subsequently married Klein. “Thewedding took place just a day after Ilearned that Vinogradov had provedthe odd Goldbach conjecture,” Erdosrecalled in 1995.

Szekeres, in a remarkable tour deforce, even rediscovered Ramsey’s the-orem, which was then only three yearsold, for his solution. This work repre-sented a milestone in Erdos’s combi-natorial thinking. Erdos recognized thevast domain opened up by Ramsey’stheorem, this “generalized pigeon holeprinciple.” In the cited 1935 paper withSzekeres, Erdos studied Ramsey num-bers for graphs, the first step in build-ing what at the hands of Erdos wouldbecome Ramsey theory, a large area infinite and transfinite combinatorics.Transfinite Ramsey theory became afundamental part of modern set theory.

One of Erdos’s heroes was GeorgCantor; Erdos learned the basics ofCantor’s set theory from his father.Erdos loved infinite cardinals and con-tributed to the birth of very large ones.(We use the term “transfinite” to em-phasize that the focus is beyond ω,usually far beyond.)

Although the methods of the finiteand the transfinite are almost disjoint(counting is fundamental in the for-mer, well-ordering in the latter), it wasErdos’s axiom that if a question makes

sense both for finite and for infinite sets, it mustbe investigated in both domains. This view is es-pecially prominent in his fifty-four (often massive)joint papers with A. Hajnal.

The chromatic number of a graph is the small-est number of colors that can be assigned to thevertices of the graph such that adjacent verticesreceive different colors. Until the mid-1950s thisconcept was mostly discussed in the limited con-text of the 4-color conjecture. Erdos’s work, whichincludes dozens of papers entitled “Chromaticgraph theory,” played a major role in establishingthe true depth of the concept. Together with Haj-

Two giants of combinatorics share a passion: Erdos and William T. Tutteplay “Go” at Tutte’s home in Westmontrose, Ontario, 1985. Another favoritegame of Erdos’s was Ping-Pong.

Erdos and Vera T. Sos in Princeton, 1985, during a period of intensivecollaboration on extremal graph theory.

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nal, Erdos also pioneered the extension of this no-tion to set systems (no member of the set systemshould be monochromatic), creating one of thepowerful unifying concepts of modern combina-torics.

One of the great early successes of Erdos’s prob-abilistic method in finite combinatorics was hisproof in 1957 that for any k, m there exists agraph of chromatic number k without cycles oflength ≤m, putting an end to a long quest by thebest combinatorial minds.

While it was relatively easy to get rid of shortodd cycles, the short even cycles proved to be muchharder to eliminate. A surprising explanation of thisphenomenon came in a milestone paper by Erdosand Hajnal, “Chromatic number of graphs and setsystems” (1966). Corollary 5.6, one of the paper’sfive dozen results, asserts: If the chromatic num-ber of a graph is ≥ ℵ1, then the graph must con-tain a 4-cycle! Buried in this paper, which is alephsall over, is an important result of finite combina-torics: Erdos’s cited result on large chromatic fi-nite graphs without short cycles is generalized toset systems.

Erdos was fascinated by the global nature of thechromatic number. A striking expression of this ishis 1962 result that to every k there is an ε > 0such that for all n > k there exist k-chromaticgraphs on n vertices such that all of their sub-graphs on ≤ εn vertices are 3-colorable. As usual,he pursued the idea for infinite graphs as well andfound that it led to a wealth of questions and sur-prising answers. With Hajnal, Erdos showed (1968)that there exist graphs of uncountable chromaticnumber on (2ℵ0 )+ vertices such that all subgraphson ≤ 2ℵ0 vertices are countably colorable. (Here α+

denotes the successor cardinal of α.) In the nicelyshaping landscape, however, as in virtually allareas of Erdos’s inquiry into the transfinite, “in-dependence raised its ugly head” after Cohen’sseminal work: a number of related questions turnedout to be independent of ZFC (Zermelo–Fraenkelset theory with the axiom of choice), even underthe Generalized Continuum Hypothesis. Amongthese is the question of the existence of an ℵ2-chro-matic graph with ℵ2 vertices such that all sub-graphs with ≤ ℵ1 vertices are ℵ0-colorable (Baum-gartner, 1984; Foreman and Laver, 1988).

If combinatorics is the art of finding patternsunder virtually no assumption, Erdos was the mas-ter of this art. Here is a simple example. A set sys-tem {A1, . . . , Ak} is called a sunflower with k petals(or a ∆-system in the original terminology of Erdosand Rado) if all pairwise intersections Ai ∩Aj areequal to

⋂ki=1Ai. Erdos and Rado recognized the

significance of this simple pattern in 1960 andshowed that for any k and r , any sufficiently largefamily of sets of size r contains a sunflower withk petals. Erdos offered $1,000 for deciding whetherCr is sufficiently large to guarantee k = 3 petals

for some (large) C . The problem remains wideopen to date. At the hands of Frankl and others,sunflowers have become a powerful tool in thestructural theory of set systems; Razborov usedthem in a profound lower-bound proof in the the-ory of Boolean circuits.

Ramsey theory provides the ultimate in thequest for simple patterns. Assume we color eachr -subset of a set S of cardinality κ red or blue. Wesay that a subset H ⊆ S is homogeneous if all r -subsets of H have the same color. The Erdos-Radosymbol κ → (α,β)r means that regardless of thecoloring, there must be either a red-homogeneoussubset of size α or a blue-homogeneous subset ofsize β . We omit β if β = α. Ramsey’s theoremstates that ℵ0 → (ℵ0)r for every finite r . Its finiteversion says that N → (k)r for sufficiently large fi-nite N = N(k, r ). The estimation of the quantitiesN(k, r ) is a major problem area. Here is an exam-ple of a tantalizing gap: it is known thatn → (c1 log logn)3 (Erdo s-Rado, 1952) andn 6→ (c2

√logn)3 (from the 100-page “giant triple

paper” by Erdos, Hajnal, and Rado (1965)).Partition calculus, the term Erdos and Rado

(1956) used for transfinite Ramsey theory, startedwith a result of Erdos that κ → (κ,ℵ0)2, includedin a 1941 paper by Dushnik and Miller. Shortly af-terwards, Erdos proved the basic result that(2λ)+ → (λ+)2 (1942) and noted that by a result ofSierpinski this bound is tight: (2λ) 6→ (λ+)2.

The fact that innocuous problems of transfinitecombinatorics lead to inaccessible cardinals was astunning discovery made in a 1943 paper by Erdosand Tarski, especially famous for its footnotes.Regarding the simplest of partition relations,κ → (κ)2, they recognized that it cannot hold un-less κ is strongly inaccessible (κ is not the sum of

Ronald L. Graham (left), Erdos, Peter Frankl finishing a paperon “anti-Ramsey graphs” at a conference in Hakone, Japan,1990. The interests shared by Graham and Frankl include

solving problems of Erdos, helping Erdos’s influence spreadin the Orient, and juggling (both are world-class jugglers).

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ano.


fewer, smaller cardinals, and α < κ implies2α < κ), and it does hold if κ is measurable (ad-mits a nontrivial < κ-additive (0,1)-measure de-fined on all subsets of κ). Out of these observa-tions, the theory of large cardinals, a vitalcomponent of modern set theory, was born. Car-dinals satisfying κ → (κ)2 are called weakly com-pact cardinals.

Another important class of large cardinals grewout of Erdos’s first joint paper with then graduatestudent András Hajnal (1958). Erdos and Hajnalproved that measurable cardinals satisfy the par-tition relation κ → (κ)<ω (all finite sets are col-ored). This relation defines what are called Ram-sey cardinals. Amazing consequences of the weakerrelation κ → (ω1)<ω to descriptive set theory werefound in the mid-1960s by F. Rowbottom, J. H. Sil-ver, R. Solovay, and others. In recognition of Erdos’spioneering role in defining large cardinals via par-tition relations, the cardinals satisfying the rela-tion κ → (ω1)<ω are commonly referred to asErdos cardinals.

We stated Ramsey’s theorem for two colors; thegeneralization to a finite number of colors is im-mediate. It is clear, however, that no homogeneoussubset is to be expected unless the set is “large”compared to the number of colors. Nevertheless,one of a short list of canonical structures exists re-gardless of the number of colors! In the simplestcase (r = 2) there are only three types of canoni-cal structure: a homogeneous set (all pairs have thesame color), a multicolored set (each pair has a dif-ferent color), and the min-coloring: the color of{i, j} is min(i, j) (assuming the set is well ordered).

This very useful fact is the Canonical Ramsey The-orem of Erdos and Rado (1950).

In a series of papers starting in 1973 Erdos,Graham, Montgomery, Rothschild, Spencer, andStraus laid the foundations of Euclidean RamseyTheory, melding Ramsey theory to the geometryof real n-space. The typical question is this: givena geometric configuration K, is it true for all r thatany r -coloring of n-space contains a monochro-matic copy of K, assuming n ≥ n0(r )? If K hasthis property, we say that K is a Ramsey configu-ration. Erdos et al. have shown that the bricks areRamsey and sets that cannot be inscribed in asphere are not Ramsey. A major problem left openin their work was settled by Frankl and Rödl in1986: all triangles are Ramsey.

Much of Erdos’s work concerns the paradigmthat “density implies pattern.” The most famous ofErdos’s solved prize problems ($1,000) assertsthat a sequence of integers of positive upper den-sity contains arbitrarily long arithmetic progres-sions. Proposed by Erdos and Turán in 1936, thisconjecture was confirmed in 1975 in “a masterpieceof combinatorial thinking” [4] by Endre Szemerédi,a disciple of Erdos and one of the most formida-ble problem solvers of our time. SubsequentlyH. Fürstenberg gave an ergodic proof. Szemerédi’sproof builds on his Regularity Lemma, which hasfar-reaching consequences in graph theory; themethod of Fürstenberg’s proof gave a new direc-tion to ergodic theory. This is but one of the longlist of examples demonstrating the profound rel-evance of the problems championed by Erdos.

A great many problems in combinatorial num-ber theory have a flavor similar to the problem ofarithmetic progressions. Young Erdos was capti-vated by Sidon’s 1934 problem, which asks howdense a set of integers can be if all pairwise sumsare different1. If we denote by A(n) the number ofelements ≤ n in such a sequence, it is clear thatA(n) ≤ c1n1/2, and “greedy” choice results in a se-quence with A(n) ≥ c2n1/3. Only very recently(1997) did Erdos protégé Imre Ruzsa succeed insubstantially reducing this sixty-year-old gap.Ruzsa has shown the existence of a Sidon sequencesuch that A(n) ≥ nα−ε with α =

√2− 1 ≈ .41.

In extremal graph theory the fundamental Erdos-Stone-Simonovits theorem (1946, 1966) considersthe minimum edge density of graphs that will

1One can hardly overestimate the influence of Sidon’sproblems on Erdos’s career. It is remarkable how thetwenty-year-old Erdos’s irresistible insistence on math-ematical communication virtually compelled Sidon, areclusive man employed by an insurance company, to re-veal his remarkable thoughts to the eager youth. A clas-sic anecdote: One afternoon, when Erdos and Turánshowed up at Sidon’s doorstep, Sidon opened the door acrack and greeted the two with these words: “Please visitanother time and especially another person.”

Paul Erdos talks about functions of a complex variable inHungary, 1959. Alfred Renyi is looking on. The landmarkErdos–Renyi papers “On the evolution of random graphs” wereborn around this time. Erdos and Renyi worked together on awide variety of other subjects as well, including thedistribution of prime numbers, complex functions, the theoryof series, probability theory, statistics, information theory, andstatistical group theory.


force the appearance of a fixed “pattern” subgraphH. It turns out that asymptotically, the density de-pends solely on the chromatic number of H! Thisin particular implies that the set of critical limit-ing densities is well ordered. Erdos asked whetherthis fact generalizes from graphs to systems of r -sets (“r -hypergraphs”) for r ≥ 3; his second (andlast) $1,000 award went to Frankl and Rödl for theirnegative answer, “Hypergraphs don’t jump” (1984).

Combinatorics and ProbabilityErdos was not versed in probability theory at thetime he arrived in Princeton in 1938. He was noteven familiar with the central limit theorem. Yethe deeply understood it in a flash when he firstheard about it in a lecture by Mark Kac; by the endof the talk he had completed the proof of Kac’s con-jecture on the normal (Gaussian) distribution of thenumber of prime divisors of integers. While thisresult gave birth to probabilistic number theory,Erdos went on and made major contributions toprobability theory itself, especially the theory ofrandom walks and Brownian motion. He workedwith Kac, K. L. Chung, Dvoretzky, Kakutani, amongothers, in these areas. Erdos’s best-known resultin probability theory is a full asymptotic expansionof the law of the iterated logarithm (1942).

A “statistical view” of mathematical objects wasone of Erdos’s key innovations in many areas ofmathematics. Following work by Goncharov in the1940s, Erdos and Turán developed statistical grouptheory, a study of the distribution of various setsof parameters associated with a group, in a seriesof seven highly technical papers between 1965and 1976. They showed, for instance, that the log-arithms of the orders of elements in the symmet-ric group Sn are asymptotically normally distrib-uted.

The foundations of a beautiful statistical the-ory of combinatorial structures were laid in thelandmark study by Erdos and Rényi of the “evo-lution of random graphs” in a series of seven pa-pers between 1959 and 1968.

Let us construct a “random” graph with n ver-tices and m edges by picking the edge set uniformly

at random from the set of (

(n2 )m

)possibilities.

Erdos and Rényi observed the typical behavior ofthese graphs as a function of m =mn and deter-mined very sharp thresholds for various mono-tone properties to become “typical.” For instance,connectedness occurs around mn = n logn/2; infact, they prove that if (mn/n)− (logn/2)→ c, thenthe probability of connectedness approaches e−e−c.

The most striking discovery of Erdos and Rényiwas a phase transition which occurs aroundmn = n/2: suddenly, a giant component appears.If mn < (1− ε)n/2, then typically all connectedcomponents of the graph are of size O(logn) andhave very simple structure. But when

mn > (1 + ε)n/2, the largest component has size> c(ε)n, while all other components remain of log-arithmic size and are absorbed into the giant com-ponent as mn increases. Béla Bollobás took the leadin a 1984 paper in uncovering the fine structureof this phase transition, the study of which hasyielded a series of remarkable insights and is con-tinuing to this day.

Much of Erdos’s work had an impact on thetheory of computing, a field in which Erdos nevertook an interest [2]. Richard M. Karp writes: “TheErdos-Rényi papers on random graphs exertedmajor influence on my work. The beautiful scenarioof the successive stages in the evolution of randomgraphs, progressing in an essentially inevitableway, has stimulated me to find other stochasticprocesses, associated with algorithms, which un-fold in the same kind of inevitability. Researchershave exhibited such processes in connection withmany problems related to graphs, Boolean for-mulas and other structures. Specific results re-lated to random graphs have been applied to hash-ing, storage allocation, load balancing and otherproblems relevant to algorithms and computersystems.”

Probabilistic Proof of ExistenceAmong the numerous techniques Paul Erdos taughtus, perhaps the probabilistic method has been themost influential. This method establishes the ex-istence of certain objects by selecting an object atrandom from a certain probability space and prov-ing that the object has the desired properties withpositive (usually overwhelming) probability. WhileErdos was not the first to employ an idea of thistype, it was he who recognized its vast scope anddeveloped it into a powerful technique.

Erdos first demonstrated the power of thismethod in 1947 by proving that his Ramsey boundwith Szekeres, n → (c logn)2 , is tight apart fromthe constant c; i.e., there exists a graph on n ver-tices without homogeneous subsets (clique or in-dependent set) of size c1 logn. The probabilitybound is obtained by generously overestimating,via simple counting, the number of graphs whichdo not have the desired property.

This non-constructive proof of existence imme-diately raised the challenge of an explicit con-struction. Frankl (1977) was the first, via the sun-flower technique, to construct explicit graphswithout homogeneous subsets of size nε ; an ele-gant alternative proof was given by Frankl andWilson using the linear algebra method (1981).Their bounds on homogeneous subsets are, how-ever, still far from logarithmic.

Another great success of the probabilisticmethod was Erdos’s cited result on the existenceof graphs with large chromatic number and with-out short cycles (1959). In this case a mere randomchoice alone will not suffice; the random graph ob-

sions of this result, which actually find “the nee-dle in the haystack,” were obtained by J. Beck andsubsequently by N. Alon (1991).

Erdos not only set up derandomization chal-lenges but also invented an important derandom-ization tool in a 1973 paper with John Selfridge:the “method of conditional expectations.” Themethod has since been extended and resulted inthe derandomization of large classes of random-ized algorithms.

For decades hardly anyone other than Erdosrecognized the significance of the probabilisticmethod. The situation changed with the 1974 pub-lication of Probabilistic Methods in Combinatoricsby Erdos and Joel Spencer. This thin volume hada major impact on all areas of discrete math-ematics.

References. We refer as [M4] to bibliographyitem [4] of the memorial article by this writer ap-pearing elsewhere in this issue. Most papers ofErdos cited above can be found either in The Artof Counting [M4] or the bibliographies of the sur-vey articles in [M12]. An enormous amount of rel-evant material appears in various chapters of themonumental Handbook of Combinatorics ([3]below). Further references:

References[1] N. ALON and J. H. SPENCER, The probabilistic method,

Wiley, New York, 1992.[2] L. BABAI, Paul Erdos (1913–1996): His influence on the

theory of computing, in Proc. 29th ACM Sympos. onTheory of Computing, ACM Press New York, 1997,pp. 383–401.

[3] R. L. GRAHAM, M. GRÖTSCHEL, L. LOVASZ, eds., Handbookof combinatorics, North-Holland, Amsterdam, 1995.

[4] R. L. GRAHAM, B. L. ROTHSCHILD, and J. H. SPENCER, Ram-sey theory, 2nd ed., Wiley, New York, 1990.

ApproximationTheoryPéter Vértesi

Paul Erdos wrote more than a hundred papers thatare closely related to the approximation of func-tions. It is difficult to outline the wealth of theseresults in such a short survey; the selection nec-essarily reflects my taste.

As a Ph.D. student of Lipót (Leopold) Fejér,Erdos began to work on problems closely relatedto interpolation and orthogonal polynomials: meanconvergence, quadrature processes, investigationsof normal point systems, and generally to revealconnections between the distribution of certain


tained from a carefully chosen distribution needsto be modified in order to satisfy the conditions.

The derandomization of this result requiredmajor effort and was eventually successful in si-multaneous work by Margulis and Lubotzky-Phillips-Sarnak (1988). A key ingredient is the the-ory of diophantine equations of the formx2 + 4q2(y2 + z2 +w2) = n (Ramanujan conjecture,solved for this case by Eichler (1954) and Igusa(1959)).

Erdos applied the probabilistic method in manyother contexts, in combinatorics as well as in num-ber theory, geometry, and analysis. Let me state abeautiful example from analysis: In a 1959 paperwith A. Dvoretzky, Erdos demonstrated the exis-tence of a power series of the form∑∞

1 eiαnn−1/2zn with real αn that diverges on thewhole unit circle.

To derandomize another probabilistic proof ofErdos in graph theory, Graham and Spencer (1971)invoked André Weil’s character sum estimates,which imply, that, in some sense, “quadraticresidues are random.” Recently (1996) Kollár,Rónyai, and Szabó employed the elements of com-mutative algebra to derandomize, for infinitelymany values of the parameters, a result of Erdosin extremal graph theory.

Our limited experience thus indicates that al-gebraic tools of considerable depth may hold thekey to replacing probabilistic proofs of existenceby explicit construction. In most cases, however,the Erdos-style proofs of existence cannot cur-rently be matched by explicit constructions, achallenge that continues to grow with the in-creasing number of applications of the proba-bilistic method [1].

Why should we care about derandomizing prob-abilistic proofs of existence? The combinatoristmay find the challenge and the beauty of the ques-tion inspiring. The reasons, however, run consid-erably deeper in the theory of computing. The cen-tral objective of that area is to show the intrinsiccomputational difficulty of explicit functions (thedifficulty of computing a random function beingevident).

The probabilistic method is most often used todemonstrate the existence of objects that are ac-tually present in abundance. For instance, a ran-dom graph is very likely to have the right Ramseyparameter. Is the method doomed to fail whensearching for rare objects? A coloring problem forset systems led Erdos and Lovász to the discoveryof the Local Lemma (1974), a powerful tool to de-tect events of low but nonzero probability. Infor-mally the lemma asserts that the intersection of aset of events, none of which correlates with morethan a small number of others, is nonempty. Thiswill demonstrate the existence of certain expo-nentially rare objects. We note that naive samplingwill not encounter these objects. Algorithmic ver-

Péter Vértesi is a senior research advisor at the Math-ematical Institute of the Hungarian Academy of Science,Budapest. His e-mail address is [email protected].


node systems and the behavior of the generatedprocesses. These questions have been in the mainstream of classical approximation theory.

However, within a very short time, Erdos beganto formulate his own problems and outlined newpaths to search, such as the closer investigationof the Lebesgue function of Lagrange interpolation,questions on the optimal Lebesgue constant, andrough and fine theories of different approximat-ing tools. The first paper in his series “Problemsand results on the theory of interpolation” ap-peared in 1958, followed by periodic updates (1961,1968, 1976, 1980, 1983, 1991).

Over the years Erdos obtained (mainly with co-authors) fundamental and very strong theorems.We may mention the 1980 result on the a.e. (almosteverywhere) divergence of Lagrange interpolationon an arbitrary system of nodes2, the Erdos con-dition on convergent interpolatory processes (withA. Kroó and J. Szabados, 1989), and the results ona.e. divergence of the arithmetic mean of Lagrangeinterpolation based on Chebyshev nodes (withG. Grünwald, 1937; an error in their proof waseliminated in a paper with G. Halász, 1991).

Questions in approximation theory are closelyrelated to the behavior of polynomials. So it is nosurprise that Erdos wrote many papers dealingwith the related problems about polynomials(Remez and other inequalities, the distribution ofroots, length of polynomials, geometry of poly-nomials, etc.). Rather than going into the detailsof this subject, I conclude this survey with two in-fluential “appetizers” and refer the interestedreader to the monographs [4, 5].

Erdos’s work invariably attracted a great deal ofattention and continues to influence the work ofmany mathematicians. This survey includes sev-eral lists of authors inspired by specific results ofErdos. References to their papers can be found inthe bibliographies of the works listed in our “Ref-erences.”

One of the most often-quoted results in ap-proximation theory appeared in a 1937 paper byErdos and Paul Turán in the Annals of Math. In thisinaugural opus of their 3-piece series “On inter-polation” the young authors proved the remark-able positive result that for any continuous func-tion f the Lagrange interpolation polynomialsconverge in mean to f if the interpolation is takenover the roots of the system of orthogonal poly-nomials with respect to any weight function. Moreprecisely they proved that for every f ∈ C andweight w

(1)∫ 1

−1{f (x)− Ln(f ,w, x)}2w (x)dx ≤

√6En−1(f ).

Here X = {xkn; 1 ≤ k ≤ n; n ∈ N} ⊂ I: = [−1,1] isan interpolatory matrix (i.e., for fixed n, xkn aredifferent); Ln(f ,X, x) ∈ Pn−1 is the n-th Lagrangeinterpolatory polynomial based on the nodes{xkn}, 1 ≤ k ≤ n ; w is a weight on [−1,1], i.e., w ≥ 0 and

∫Iw > 0 ; if {xkn = xkn(w )} where

xkn(w ), 1 ≤ k ≤ n are the roots of the n-th ortho-normal polynomial (pn(w ) ) with respect to w , n ∈ N (i.e.,

∫Ipn(w )pm(w )w = δnm), then Ln(f ,w )

replaces Ln(f ,X) ; finally, En(f ) = minP∈Pn ‖f − P‖where ‖ . . .‖ stands for the maximum norm on I.Note that by Weierstrass’s Theorem, the right-hand side converges to 0 as n →∞.

To appreciate this mean-convergence theorem,we state a fundamental negative result of G. Faber(1914), which says that for every X ⊂ I there is anf ∈ C with

(2) lim supn→∞

‖Ln(f ,X, x)‖ =∞.

A natural question that challenged many math-ematicians was to replace the exponent 2 with alarger one. Such results are known for special ma-trices. For instance, for the case ofX = T =

{cos 2k−1

2n π}

(Chebyshev matrix) Erdosand Feldheim proved in 1936 that

Erdos enjoyed working with several mathematicians onentirely different problems simultaneously. Left to right:

G. Grätzer, Erdos, Paul Turán, and Alfréd Rényi at Dobogóko ,Hungary, 1959. Turán was one of Erdos’s closest friends and

his first major collaborator. Erdos and Turán worked togetheron a variety of subjects in number theory, classical analysis,

combinatorics, and statistical group theory.

2Note by the organizer: The coauthor of this striking re-sult is P. Vértesi.


limn→∞

∫ 1

−1|f (x)− Ln(f , T , x)|p 1√

1− x2dx = 0

holds for any continuous f with arbitrary p > 0.The corresponding trigonometric case is due toJ. Marcinkiewicz. As it turned out almost forty (!)years later, however, generally the exponent 2 can-not be improved. This nice result is due to P. Nevai.Similar problems were considered by, among oth-ers, R. Askey, V. M. Badkov, B. Della Vecchia,G. Freud, G. Mastroianni, B. Muckenhoupt, D. S. Lu-binsky, A. K. Varma, and Y. Xu (cf. [2]).

Lebesgue estimated the difference Ln(f )− f by

(3) |Ln(f ,X, x)− f (x)| ≤ {λn(X,x) + 1}En−1(f ).

Here the n-th Lebesgue function λn(X,x) is de-fined as λn(X,x): =

∑nk=1 |`kn(X,x)| , where the

`kn ∈ Pn−1 \ Pn−2 are the (unique) fundamentalpolynomials corresponding to X (i .e. ,`kn(X,xjn) = δkj , 1 ≤ k, j ≤ n , n ∈ N). Relation(3) shows that the Lebesgue function λn(X,x) andthe Lebesgue constant Λn(X): = ‖λn(X,x)‖ play afundamental role concerning the convergence-di-vergence behavior of Lagrange interpolation.

In the seminal first paper in the “Problems …”series (1958), Erdos proved that for any fixedX ⊂ [−1,1], real ε > 0, A > 0, the measure of theset for which

(4) λn(X,x) ≤ A, x ∈ R, n ≥ n0(A, ε),

is less than ε.The basic ideas of this work were used, devel-

oped, and completed by Erdos and many (co)au-thors (G. Halász, D. Newman, J. Knoppenberger,J. Szabados, A. K. Varma, P. Vértesi, Y. G. Shi) in aseries of papers. These papers resulted in more orless “best possible” theorems on the behavior ofλn(X,x) and similar expressions, and they gave far-reaching generalizations of the Faber theorem andthe Grünwald-Marcinkiewicz result (see the pa-pers highlighted in the fourth paragraph of this sur-vey and further references in [1]).

It is natural to investigate the sequence

(5) Λ∗n : = minX⊂I

Λn(X), n ∈ N.

In the rather difficult second paper of the “Prob-lems …” series (1961) Erdos, improving on a jointresult with P. Turán (1961), obtained the bound

(6) |Λ∗n − 2π

logn| ≤ c, n ≥ n0,

but the famous Bernstein-Erdos conjectures onthe optimal matrix X∗ for which Λn(X∗) = Λ∗n andon the behavior of λn(X∗, x) were proved only in1978 (T. Kilgore, C. deBoor, A. Pinkus, L. Brutman

(cf. [1])). The bound (6) also attracted much in-terest; a long list of papers on this subject is citedin [1].

Let us now consider functions f satisfying theLipschitz condition |f (x)− f (y)| ≤ c|x− y|α withsome constant c; Lip(α) denotes the class of suchfunctions. For f ∈ Lip(α), 0 < α < 1, (3) yields thebound

(7) ‖Ln(f ,X)− f‖ ≤ cn−αΛn(X).

In their 1955 joint paper “On the role of theLebesgue function in the theory of Lagrange in-terpolation” Erdos and Turán established the fol-lowing surprising facts.

Let us suppose that Λn(X) ∼ nβ (β > 0). Then ifα > β , we have uniform convergence for anyf ∈ Lip(α); if α < β

β+2, then for some f1 ∈ Lip(α),‖Ln(f1, X)‖ is unbounded as n →∞. However, ifββ+2 < α < β , then both convergence and diver-gence can happen.

This means that in the third case the conver-gence-divergence behavior of Ln(f ,X) is not de-termined by the order of Λn(X) alone; we have totake a closer look at the matrix X itself. Erdos andTurán refer to the interval [β/(β + 2), β] as the do-main of a finer theory and point out a number ofanalogous situations for further study.

This is an extremely influential work. Over theyears dozens of papers tried to settle corre-sponding questions (rough and fine theory) forthe trigonometric case, other operators, Hermite-Fejér interpolation, etc. (cf. [1]).

Now here are two results on polynomials. Ac-cording to the Bernstein-Markov inequality,

(8)|p′n(x)| ≤min

(n√

1− x2, n2

)· ‖pn‖,

|x| ≤ 1, pn ∈ Pn.

However, as Erdos has shown in a short paper(1940), we can do better if we restrict the zeros ofthe polynomial. Namely, if pn ∈ Pn has no root in(−1,1), then

(9)|p′n(x)| ≤min

(4√n

(1− x2)2,en2

)· ‖pn‖,

|x| ≤ 1.

This result was one of the starting points of in-vestigations on polynomials with restricted zerosand initiated many interesting general problems(cf. the works of P. Borwein, T. Erdélyi, M. vonGolitschek, G. G. Lorentz, Y. Makovoz, A. Máté,J. Szabados, A. K. Varma, and others mentioned inthe monograph [3]).

Let me close this survey with comments on an-other short paper by Erdos, “On the distribution


of roots of orthogonal polynomials.” In this 1972paper, which represents a bridge between ap-proximation theory and polynomials, Erdosshowed that certain weights w with infinite sup-port have the so-called arcsine distribution; i.e.,the distribution of the contracted zeros of thecorresponding orthogonal polynomials is simi-lar to the root-distribution of the Chebyshevpolynomials (“Erdos-type weights,” as they arereferred to today3).

During the last fifteen to twenty years inves-tigations of so-called weighted approximationson R (approximating f (x)w (x) by pn(x)w (x) ,pn ∈ Pn, x ∈ R) have been very intensive; manyapproximating tools and formulae were devel-oped (mainly for Erdos- and Freud-type weights)by G. Freud, A. Levin, D. S. Lubinsky, H. N.Mhaskar, P. Nevai, E. A. Rahmanov, E. B. Saff,J. L. Ullman, V. Totik, R. S. Varga, and their stu-dents (cf. the monograph [5] and the referencestherein). The next stage should be the investi-gation of the previously mentioned problemsconcerning weighted interpolation. The firststeps have already been done; they clearly markthat Paul Erdos’s ideas are very much alive.

References[1] J. SZABADOS and P. VÉRTESI, Interpolation of functions,

World Sci. Publ., London and Singapore, 1990.[2] ———, A survey on mean convergence of interpo-

latory processes, J. Comput. Appl. Math. 43 (1992),3–18.

[3] P. BORWEIN and T. ERDÉLYI, Polynomials and poly-nomial inequalities, Springer-Verlag, Berlin andNew York, 1995.

[4] G. G. LORENTZ, M. VON GOLITSCHEK, and Y. MAKOVOZ,Constructive approximation: Advanced problems,Springer-Verlag, Berlin and New York, 1996.

[5] A. A. LEVIN and D. S. LUBINSKY, Christoffel functionsand orthogonal polynomials for exponentialweights on [-1,1], Mem. Amer. Math. Soc. 535(1994), 1–145.

[6] P. TURAN, Collected papers of Paul Turán, P. Erdos,ed., Akadémiai Kiadó, Budapest, 1990.

3Let w (x) = e−Q(x). If Q(x) = |x|α, α > 1, x ∈ R (Freud-weight), then w is not arcsine; on the other hand forQk(x) = exp(exp(. . . exp(|x|α) . . .)) (k ≥ 1 times), α > 0,x ∈ R (Erdos-weight), w is arcsine (cf. Erdos-Freud(1974)).


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