?W H A T I S . . .

a Flag Algebra?Alexander A. Razborov

Before attempting to answer the question from thetitle, it would be useful to say a few words aboutanother question: what kind of problems have flagalgebras been invented for?

Let us consider three similar combinatorialpuzzles. Assume that we have a (simple, undirected)graph with n vertices. What is the minimal numberof edges m (as a function in n) that guaranteesthe existence of a triangle? Also, assuming that mis above this threshold, how many triangles areguaranteed to exist? Let us now offset everythingby one, and instead of graphs consider (simple)3-graphs, i.e., sets of unordered triples (called3-edges) on n vertices. We again ask, what is theminimal number m of 3-edges that guaranteesthe existence of four vertices such that all fourpossible triples spanned by these vertices are inthe set of 3-edges?

The subarea of discrete mathematics that dealswith questions of this sort is called extremalcombinatorics, and it is very strategically located ata crossroads between “pure” mathematics and itsapplications. One good way to describe flag algebrasis as an attempt to expose and emphasize somecommon mathematical structure underlying manystandard techniques in extremal combinatorics,and a survey of concrete results obtained in thisway can be found in [1]. Before going into more

Alexander A. Razborov is the Andrew MacLeish Distin-guished Service Professor in the Department of ComputerScience at the University of Chicago. Part of this work wasdone while the author was at the Steklov MathematicalInstitute, supported by the Russian Foundation for Basic Re-search, and at Toyota Technological Institute, Chicago. Hisemail address is razborov@cs.uchicago.edu.

DOI: http://dx.doi.org/10.1090/noti1051

detail, however, let me encourage the reader toput this article aside and try to predict the currentstatus of the three problems from the previousparagraph.

Ready? The first problem (on the thresholdvalue m(n)) was solved in a classic paper byMantel published in 1907. The second problem(on the minimal number of triangles beyond thethreshold) had been open for some forty years.It was asymptotically solved only recently usingflag algebras. The generalization to 3-graphs wassuggested by Turán in another classic paper written

in 1941; he conjectured thatm ≤ ( 59+o(1))

(n3

)and

gave the first construction attaining this bound.This conjecture remains unsolved despite repeatedattempts by many strong researchers, and it hasgreatly stimulated the development of the wholefield. Some partial results toward Turán’s (3,4)-conjecture (by the way, its analogues are open forany values of the parameters 4 and 3 as long as4 > 3 > 2), though, were obtained with the helpof flag algebras. As Sidorenko, one of the leadingexperts in the area, put it in his survey dated 1995,“The general problem of Turán having an extremelysimple formulation but being extremely hard tosolve, has become one of the most fascinatingextremal problems in combinatorics.”

Now, let us do one of the many proofs of Mantel’sresult: it will serve as a motivating running examplefor our definitions. Let d1, . . . , dn be vertex degreesin our graph, and let e(v) ≈ dv/n be the relativedegree of the vertex v ∈ {1,2, . . . , n}. (Strictlyspeaking, e(v) = dv/(n − 1), but systematicallyignoring low-order terms is one of the most basicprinciples of the theory we are discussing.) We

1324 Notices of the AMS Volume 60, Number 10

have(1)

0≤ 1n

n∑v=1

(e(v)− 12)2≈ 1

n3

∑vd(v)2− 1

n2

∑vd(v)+1

4.

The term (1/n2)∑v d(v) is easy to interpret: it

is simply (remember that we are ignoring low-order

terms!) the edge density ρ (=m/(n2

)). To calculate

(1/n3)∑v d(v)2, we use double counting and we

look at configurations in our graph spanned bythree vertices. We can see there are 0, 1, 2, or3 edges, and let us denote by I3, P3, P3, K3 therespective densities or probabilities with whichthese configurations occur (“I” stands for “inde-pendent”, “P” stands for “path”, “P” stands for“complement of a path”, and “K” stands for amisspelled “clique”). Then a moment’s reflectionreveals that (1/n3)

∑v d(v)2 ≈ (1/3)P3 +K3: only

these two cases contribute to the sum, and K3

contributes thrice as much since we have threedifferent choices of v in it. But ρ can also beexpressed in these terms as ρ = 1

3 P3 + 23P3 + K3:

we generate a random pair of vertices in two steps,first by picking a random triple and then by select-ing a random pair within this triple. Plugging allthis into (1), after simple calculations we concludethat ρ ≤ (1/2)− P3+K3. Thus, ρ > 1/2 implies theexistence of triangles, and, moreover, their densityK3 satisfies K3 ≥ ρ − 1/2. If we are slightly morecareful and instead of (1/n) ·

∑nv=1(e(v)− 1/2)2

compute (1/n)·∑nv=1(e(v)−ρ)2 (that is, the actual

variance of the degree sequence), we will get abetter bound K3 ≥ ρ(2ρ − 1) proved by Goodmanin 1959. This latter bound had remained the bestknown for the second problem on our list until itwas superseded in a beautiful paper by Bollobás(1975).

Let us now see which kind of structure wecan extract from this template (all this materialcan be found in [2]). Our target graph G is largeand unknown. Thus, for every fixed graph H weintroduce a formal real-valued variable with thesame name and the intuitive meaning “the densityof induced copies of H in G” (in the argumentabove, H was one of ρ, I3, P3, P3, or K3). As weoften need to sum these quantities with realcoefficients, we form the linear space of formal(finite) linear combinations of these variables. Theidentity ρ = 1

3 P3 + 23P3 + K3 that we used above

can be widely generalized: the density of any fixedgraph H can be expressed in terms of densitiesof graphs with a fixed but larger number ofvertices. We factor our space out by these relations.Multiplication is also available: for example, ρ2

can be expressed as a linear combination of [thedensities of] graphs on four vertices; this is doneby double counting similar to our calculation of

(1/n3)∑v d(v)2. All these developments give us a

commutative associative algebraA0.Furthermore, when the size of the target graphG

tends to infinity, the densities of induced copies ofH converge to an algebra homomorphism φ fromA0 to R. These algebra homomorphisms possessan extra property that φ(H), being a density, isnonnegative for any graph H. Then it turns outthat we also have an important “completenessresult”: every abstract algebra homomorphismfromA0 to R with this nonnegativity property (letus call their set Hom+(A0,R)) can be obtainedfrom a convergent sequence {Gn}. In other words,the object Hom+(A0,R) defined in the best math-ematical traditions quite abstractly nonethelesscorresponds exactly to the class of extremal prob-lems we intend to study. For example, the secondproblem on our list can be reformulated like this:given x ∈ [0,1], compute the minimal possiblevalue of φ(K3), where φ ∈ Hom+(A0,R) satisfiesφ(ρ) = x, and, yes, we do mean min here, not inf,since Hom+(A0,R) is compact. The computationdoes begin with the words “let us fix once and forall an extremal φ.”

Having thus reformulated the questions of studyin the appropriate language, the rest of the theoryis basically devoted to developing useful syntactictools for proving theorems about the behaviorof the limit densities φ(H). Most of these toolshave evolved from analogous methods employedin finite arguments, but, again, the mathematicalstructure allows us to search for the desired proofseither completely automatically or in an interactivecomputer-human mode. This allows us to expandthe search space by an order of (literally!) hundredsor thousands. We once more refer to [1] for asurvey of concrete results that have been obtainedwith this method, and now we review some ofthese tools.

Our account above was tailored to the case ofordinary graphs, but this was done only for simplic-ity of exposition. The theory of flag algebras wasdeliberately set up in such a way that it applies in auniform way to arbitrary combinatorial structures:hypergraphs, directed graphs, mixtures of these,colored versions of these, you name it. In logicalterms, the set-up can be described as a “universaltheory in a language containing only predicatesymbols,” but the only property that is actuallyneeded is that a subset of vertices of a model spansan (induced) submodel. This is paramount sincestructures other than simple graphs are where themost important open problems in the area reside,Turán’s (3,4)-problem being just the tip of theiceberg.

Next, our definitions can be readily generalizedto the case when all our models are required tocontain k distinguished “base” vertices (for an

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analogy, the reader should think of base points inalgebraic topology) that must be preserved by allmapping involved. This clearly makes sense onlywhen we also specify what we see on c1, . . . , ckitself; for example, if k = 2, then in the graph theorywe should specify whether (c1, c2) is an edge ornot. Thus, we say that a type σ is simply a modelwith the ground set {1,2, . . . , k}, a flag1 of typeσ is a model with k base vertices respecting thestructure of σ , and then (finally!) the flag algebraAσ and Hom+(Aσ ,R) are defined just as before.As an example, let 1 be the (only) type of size 1,and let e ∈A1 be an edge with one distinguishedvertex in it. Then, instead of computing the sum(1/n3)

∑nv=1 d(v)2 in (1), we could alternatively

express first e2 = K13 + P

1,c3 , where K1

3 and P1,c3 are

1-flags that are obtained from K3, P3, respectively,by adding one base vertex which is the center ofthe path in the case of P3.

Every flag algebra Aσ has a good supply ofelements that are guaranteed to be evaluated toa nonnegative value by any φ ∈ Hom+(Aσ ,R):these are the squares f 2 and their positive linearcombinations. We can also define a linear averagingoperator J·Kσ that generalizes the summation in(1). For example, JK1

3K1 = K3, JP1,c3 K1 = (1/3)P3,

which gives us another proof of what we did abovedifferently, Je2K1 = (1/3)P3 + K3. Now we have ahealthy supply of nonnegative elements inA0, too:these are Jf 2Kσ , where f ∈ Aσ for some type σ ,and their positive linear combinations, possiblymixing up relations coming from different types.

It already turns out that these simple ideas(called in [1] “plain methods”) can solve many openproblems if you look deep enough, i.e., for f 2 ∈Aσ

that involve flags on sufficiently many vertices(from four to six in a typical application). It isobvious from the look of most of these results thatthey could hardly be obtained by hand and thatthey do require computer assistance. Fortunately,the question of “how to represent a specific f in theform

∑i αiJf 2

i Kσi (αi ≥ 0)” is completely answeredby semidefinite programming (SDP), and, equallyfortunately for flag algebras, for the latter wehave not only theoretical results (polynomial timealgorithms) but also good noncommercial packagesthat really do the job. In my own work I most oftenuse CSDP, a package developed by Brian Borchers

1The choice of the term “flag” to stand for “a partially la-beled combinatorial structure in which labeled vertices spana prescribed model σ” is admittedly somewhat arbitrary. Itis largely suggested by a visual association: a few verticesare fixed rigidly while many more are “free” and “waving”through the model we are studying. It has very little to dowith other usages of this term in mathematics…incidentally,I have never seen a good explanation of what increasingsequences of linear spaces have to do with corporeal flags,either.

(https://projects.coin-or.org/Csdp); it isalso used in the publicly available flagmaticsoftware (http://www.maths.qmul.ac.uk/~ev/flagmatic/) by Emil R. Vaughan. One thingthat greatly hinders these developments is theabsence (to the best of my knowledge) of genericSDP-solvers that also provide certificates offeasibility/unfeasibility: many actual calculationsin flag algebras are performed so close to theborder between them that numerical results oftencannot be trusted.

Besides purely computational convenience, flagalgebras lead to more sophisticated structuresand objects that also have found applications inconcrete proofs; due to lack of space we can onlyname some of them here (see [2] for all missingdetails).

There are many useful constructions that allowus to convert combinatorial objects of one sortinto objects of another sort. For example, given adirected graph, we can view it as an ordinary graphby erasing its orientation, or, given a 3-graph anda vertex in it, we can look at its link, which is againan ordinary graph. Constructions of this kind arecaptured by the logical notion of an interpretation,and in the language of flag algebras they lead toalgebra homomorphisms betweenAσ for differenttheories. This allows us to conveniently movearound theorems (i.e., statements of the formf ≥ 0) from one context to another.

The space Hom+(A0,R) is compact, which im-plies that, for every extremal problem, there existsan individual optimal solution to it. Extremalitycan be exploited in a variety of ways: for exam-ple, one can write (an analogue of) a functionalderivative according to some intuitive changesand use its equality to zero as a new usefulrelation not necessarily possessed by an arbitraryφ ∈ Hom+(A0,R).

Tuples of densities φ ∈ Hom+(A0,R) encodea surprising amount of information about theintended object. For example, if we “pick” abase vertex (or, more generally, a copy of amore complicated type σ ) at random, then it“should” give rise to a probability distributionover φ ∈ Hom+(Aσ ,R). It turns out that thisinformation can already be uniquely retrievedfrom φ even when we do not really have a goodidea which space we are sampling from. Forexample, we can straightforwardly (say, avoidingSzemeredi’s Regularity Lemma) determine thefraction of vertices in a graph that have relativedegree at least 1/3 using φ only.

It is tempting to employ a similar axiomaticapproach in other areas where the Cauchy-Schwarzinequality is used. The difficulties arising here areof a more technical and practical nature: sincethe group of symmetries is not as rich in those

1326 Notices of the AMS Volume 60, Number 10

situations, it is more difficult to come up with acalculus that is good not only theoretically butalso allows us to get new concrete results. Somework in this direction, however, has already beendone; see [1, Section 4.2].

Last, but definitely not least, we are ofteninterested not only in the properties of the limitdensitiesφ itself, but also in what is the actual limitobject these densities correspond to (or, in morelogical terms, in the associated model theory). Thisleads to the deep and beautiful theory of graphlimits with many connections to other disciplines,and we highly recommend Lovász’s recent book [3]for an introduction to the subject. We strongly feelthat emphasizing more connections between thesyntactical (flag algebras) and semantical (graphlimits) approaches to the same class of objectsshould be very beneficial for both.

References[1] A. Razborov, Flag algebras: An interim report, 2013,

to appear in the Erdos Centennial Volume, prelimi-nary text available at http://people.cs.uchicago.edu/~razborov/files/flag_survey.pdf.

[2] , Flag algebras, Journal of Symbolic Logic,72(4):1239–1282, 2007.

[3] L. Lovász, Large Networks and Graph Limits, AmericanMathematical Society, 2012.

November 2013 Notices of the AMS 1327