The Random Graph (Universal Structures II)

In our previous post, we provided a literary prologue to the study of universal structures, objects that, in a sense, “contain” all other objects of the same type. Today, we return to the world of mathematics to look at a fascinating concrete example of such a universal structure.

The mathematical objects we will be working with are graphs, which we briefly introduced in our Ultrafilters series. Recall that a graph is a set of vertices and a set of edges, where each edge connects two distinct vertices. For example:

graph1

A graph G is often represented as a pair (V,E), where V is the set of vertices and E is the set of edges. An edge is represented by a set containing the two vertices it connects. If G_1 = (V_1, E_1) and G_2 = (V_2, E_2) are two graphs, then we say that G_1 embeds into G_2 if there is a function f:V_1 \rightarrow V_2 such that:

  • for all u,v \in V_1, if u \neq v, then f(u) \neq f(v);
  • for all u,v \in V_1, if \{u,v\} \in E_1, then \{f(u), f(v)\} \in E_2;
  • for all u,v \in V_1, if \{f(u), f(v)\} \in E_2, then \{u,v\} \in E_1;

One can think of the statement “G_1 embeds into G_2” as saying that “G_2 contains a copy of G_1.” To really understand this idea, try the following exercise.

Exercise: Every graph with three vertices embeds into the graph pictured above.

In light of this exercise, we might be tempted to say that the graph we have drawn above is universal for graphs with three vertices. However, we are cheating a little bit, since our graph has more than three vertices itself. We would really like for a universal object to itself be a member of the class for which it is universal; otherwise, finding universal objects is too easy! A bit of thought should convince you that, with this more stringent definition of “universal,” no graph that is universal for a meaningful class of graphs can be finite (the word “meaningful” here is just meant to rule out whatever trivial examples you can think of).

We therefore will waste no time trying to find finite universal graphs and will instead move on to the next step: graphs with a countable infinity of vertices. And here we will succeed: we will find a graph G with countably many vertices such that, for every graph H with countably many vertices, H embeds into G. What’s more, we will succeed with a wonderfully interesting graph: the random graph.

One commonly cited criterion for an idea or an object to possess mathematical beauty is that it arise independently in a variety of different contexts, in a variety of guises that are only revealed to be identical upon close inspection. By this accounting, I would say that the random graph is quite beautiful indeed. Let us give just a few descriptions of this graph.

1: Hereditarily finite setsFor a set x, the transitive closure of x is, roughly, the set consisting of all elements of x, all elements of elements of x, all elements of elements of elements of x, and so on. It is quite possible that a set x is finite but its transitive closure is infinite (consider, for example, the set whose one element is the set of all natural numbers). A set is called hereditarily finite if its transitive closure is finite. Assuming that all sets are built up from the empty set, one can prove that there are only countably many hereditarily finite sets. Moreover, if n is a natural numbers and x_1, x_2, \ldots, x_n are hereditarily finite sets, then \{x_1, x_2, \ldots, x_n\} is a hereditarily finite set. Now form a graph G = (V,E) by letting V be the set of all hereditarily finite sets and, for all x,y \in V, letting \{x,y\} \in E if and only if either x \in y or y \in x.

2: Binary expansion. We typically write natural numbers in base 10, or decimal, notation, but one could just as well write them in base 2, or binary, notation, using only 0s and 1s. I will assume the reader is at least passingly familiar with this. As a brief refresher, recall that, in binary notation, 1 is 1, 2 is 10, 3 is 11, 4 is 100, 5 is 101, and so on. We define a graph G = (V,E) such that V is the set of natural numbers. If m < n are natural numbers, then we put an edge between m and n if and only if the m^{\mathrm{th}} digit (from the right, starting with 0) in the binary expansion of n is 1. For example, the number 10 in binary is 1010. Therefore, there is no edge between 0 and 10 or 2 and 10 but there is an edge between 1 and 10 and between 3 and 10. If 4 \leq m < 10, then there is no edge between m and 10 because the binary expansion of 10 only has 4 digits.

Rado_graph
The first nine vertices of the binary expansion graph. Illustration by David Eppstein.

3: Coin flipping. We will define a graph G = (V,E) probabilistically, where V is the set of natural numbers. For each pair of natural numbers m < n, we will flip a fair coin. If the coin lands heads, we put an edge between m and n. If it lands tails, we do not put an edge between m and n.

4: Quadratic residues. If p and q are natural numbers, we say that p is a quadratic residue modulo q if there is a natural number x such that x^2 - p is divisible by q. We define a graph G = (V,E) such that V is the set of all prime numbers that leave a remainder of 1 when divided by 4. It is an elementary fact of number theory that there are infinitely many such prime numbers. If p < q are two elements of V, then we put an edge between p and q if and only if p is a quadratic residue modulo q or q is a quadratic residue modulo p.

5: Fraïssé limit. Let G be the Fraïssé limit of the class of finite graphs. I am not going to explain this further; I just wanted to use the phrase “Fraïssé limit.”

One of the almost miraculous aspects of our story is that all five of these graphs are the same, and we shall refer to all of them as “the random graph.” Before we go any further, though, I should explain what I mean by “the same,” since these graphs are not literally “the same” (some of them don’t even have the same sets of vertices). By “the same” we mean isomorphic, or “structurally the same,” in the way in which we would say that any two graphs with three vertices and edges between all three vertices are “the same.” More precisely, if G_1 = (V_1, E_1) and G_2 = (V_2, E_2) we say that G_1 and G_2 are “the same” (or isomorphic) if there is a one-to-one correspondence (i.e., a bijection) f:V_1 \rightarrow V_2 such that, for all x,y \in V_1, we have \{x,y\} \in E_1 if and only if \{f(x), f(y)\} \in E_2. The point is that the way in which we label the vertices of a graph is immaterial to the actual structural properties of the graph, which are what we care about.

Now, how can we show that all of these graphs are the same? We show that all of the graphs have a very special property, and that all countable graphs with this property are isomorphic. This property is called the extension property.

Definition: A graph G = (V,E) satisfies the extension property if, for all finite subsets U_1, U_2 \subseteq V with U_1 \cap U_2 = \emptyset, there is a vertex v \in V such that:

  • v \not\in U_1 \cup U_2;
  • for all u \in U_1, we have \{u,v\} \in E;
  • for all u \in U_2, we have \{u,v\} \not\in E.

I will leave the verification that all five of our graphs have the extension property as an exercise. For graphs 1 and 2, this is pretty easy. For graph 3, since the definition of the graph is probabilistic, it is conceivable (just ask Rosencrantz and Guildenstern) that every coin flip somehow comes up heads, in which case the graph would obviously not satisfy the extension property. However, the graph will satisfy the extension property with probability 1, so we can be almost sure. The fact that graph 4 satisfies the extension property follows from the Chinese Remainder Theorem and Dirichlet’s theorem about primes in arithmetic progressions, two somewhat non-trivial theorems from number theory.

Theorem: Suppose G_1 = (V_1, E_1) and G_2 = (V_2, E_2) are two countably infinite graphs with the extension property. Then G_1 and G_2 are isomorphic.

Proof sketch: Since G_1 and G_2 are countable, we can enumerate V_1 as \{u_0, u_1, u_2, \ldots \} and V_2 as \{v_0, v_1, v_2, \ldots \}. We will construct an isomorphism f:V_1 \rightarrow V_2 in infinitely many steps by defining f on more and more elements of V_1.

Suppose n is a natural number. We now describe Step n of the construction. Assume that we have already defined f on finitely many elements of V_1 in such a way that f is thus far an isomorphism, i.e., for all pairs u \neq v on which f is thus far defined, we have f(u) \neq f(v) and \{u,v\} \in E_1 if and only if \{f(u), f(v)\} \in E_2. Suppose moreover that, for all m < n, f is defined on u_m and there is u \in V_1 such that f is defined on u and f(u) = v_m.

We now extend f so that it is defined on u_n. If it happens to be the case that it is already defined on u_n, then we have nothing to do. Otherwise, let U be the set of vertices on which f is thus far defined. Let U_1 be the set of vertices in U to which u_n is connected by an edge, and let U_2 be the set of vertices in U to which u_n is not connected. Let W_1 = \{f(u) \mid u \in U_1\} and W_2 = \{f(u) \mid u \in U_2\}. Since G_2 satisfies the extension property, we can find v \in V_2 such that:

  • v \not\in W_1 \cup W_2;
  • for all w \in W_1, we have \{w,v\} \in E_2;
  • for all w \in W_2, we have \{w,v\} \not\in E_2.

Let f(u_n) = v. Next, we extend f so that v_n is in its image. If there is already u \in V_1 such that f(u) = v_n, then there is nothing to do. Otherwise, let W be the set of w \in V_2 such that, for some u \in V_1, we have f(u) = w. Let W_1 = \{w \in W \mid \{v_n, w\} \in E\} and W_2 = \{w \in W \mid \{v_n, w\} \not\in E\}. Let U_1 = \{u \in V_1 \mid f(u) \in W_1\} and U_2 = \{u \in V_1 \mid f(u) \in W_2\}. Since G_1 satisfies the extension property, we can find u \in V_1 such that:

  • u \not\in U_1 \cup U_2;
  • for all v \in U_1, we have \{u,v\} \in E_1;
  • for all v \in U_2, we have \{u,v\} \not\in E_1.

Let f(u) = v_n. This completes Step n of the construction. Continuing through all of the steps, one for each natural number, we find that we have constructed a full isomorphism from G_1 to G_2, thus completing the proof of the theorem.

The theorem shows that we are justified in referring to the random graph rather than random graph. Up to isomorphism (and with probability 1), all of the graphs we have called “the random graph” are the same! Essentially the same proof as that given above shows that every countable graph embeds into the random graph (Exercise: Do it!). Thus, the random graph can truly be said to be universal for the class of countable graphs.

P.S. To what extent do these ideas generalize to higher infinities? Assuming some assumptions about cardinal arithmetic, they generalize quite straightforwardly. If \kappa is an infinite cardinal and \kappa^{<\kappa} = \kappa, then it is meaningful to talk about the “random graph of size \kappa,” and this graph will be universal for graphs of size \kappa. However, the assumption \kappa^{<\kappa} = \kappa is not necessarily true for uncountable cardinals. In fact, it is consistent with the axioms of set theory that there are no universal graphs of size \aleph_1. The question of the existence of universal graphs becomes especially interesting at cardinals of the form \mu^+, where \mu is a singular cardinal; much work has been done even in the last few years on this topic.

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The Simurgh (Universal Structures I)

To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour

-William Blake, “Auguries of Innocence”

In Persian mythology, the Simurgh is a bird that lives in the mountains of Alborz. Sometimes she has the head or body of a dog, sometimes of a human. She has witnessed the destruction of the world three times. The wind of her beating wings is responsible for scattering seeds from the Tree of Life, creating all plants in the world.

The Simurgh is, in some tellings, the archetype of all birds. Her name resembles the Persian phrase si murg, meaning “thirty birds.”

In The Conference of the Birds, Farid ud-Din Attar’s 12th-century masterpiece, the birds of the world undertake a journey to find the Simurgh. And they succeed.

Their life came from that close, insistent sun
And in its vivid rays they shone as one.
There in the Simorgh’s radiant face they saw
Themselves, the Simorgh of the world – with awe
They gazed, and dared at last to comprehend
They were the Simorgh and the journey’s end.
They see the Simorgh – at themselves they stare,
And see a second Simorgh standing there;
They look at both and see the two are one,
That this is that, that this, the goal is won.

-Farid ud-Din Attar, The Conference of the Birds

The Simurgh is a bird that contains all birds. She is a universal bird.

Unsurprisingly, the Simurgh shows up a number of times in the works of Jorge Luis Borges, in both his short stories and his essays. One reference appears in the masterful story, “The Aleph,” a particularly rich and dense work which you should certainly read for yourself.

“The Aleph” is partly about how we create our own worlds, how we approximate the unknowable universe within our lives and our art. The narrator of the story, also named Borges, grieving the loss of his beloved Beatriz, pays repeated visits to the home of her father and her cousin, the poet Carlos Argentino Daneri. On one of these visits, Carlos Argentino takes Borges to his basement to show him the source of his poetry, the titular Aleph, a single point that contains the universe.

On the back part of the step, toward the right, I saw a small iridescent sphere of almost unbearable brilliance. At first I thought it was revolving; then I realised that this movement was an illusion created by the dizzying world it bounded. The Aleph’s diameter was probably little more than an inch, but all space was there, actual and undiminished. Each thing (a mirror’s face, let us say) was infinite things, since I distinctly saw it from every angle of the universe. I saw the teeming sea; I saw daybreak and nightfall; I saw the multitudes of America; I saw a silvery cobweb in the center of a black pyramid; I saw a splintered labyrinth (it was London); I saw, close up, unending eyes watching themselves in me as in a mirror; I saw all the mirrors on earth and none of them reflected me; I saw in a backyard of Soler Street the same tiles that thirty years before I’d seen in the entrance of a house in Fray Bentos; I saw bunches of grapes, snow, tobacco, lodes of metal, steam; I saw convex equatorial deserts and each one of their grains of sand; I saw a woman in Inverness whom I shall never forget; I saw her tangled hair, her tall figure, I saw the cancer in her breast; I saw a ring of baked mud in a sidewalk, where before there had been a tree; I saw a summer house in Adrogué and a copy of the first English translation of Pliny — Philemon Holland’s — and all at the same time saw each letter on each page (as a boy, I used to marvel that the letters in a closed book did not get scrambled and lost overnight); I saw a sunset in Querétaro that seemed to reflect the colour of a rose in Bengal; I saw my empty bedroom; I saw in a closet in Alkmaar a terrestrial globe between two mirrors that multiplied it endlessly; I saw horses with flowing manes on a shore of the Caspian Sea at dawn; I saw the delicate bone structure of a hand; I saw the survivors of a battle sending out picture postcards; I saw in a showcase in Mirzapur a pack of Spanish playing cards; I saw the slanting shadows of ferns on a greenhouse floor; I saw tigers, pistons, bison, tides, and armies; I saw all the ants on the planet; I saw a Persian astrolabe; I saw in the drawer of a writing table (and the handwriting made me tremble) unbelievable, obscene, detailed letters, which Beatriz had written to Carlos Argentino; I saw a monument I worshipped in the Chacarita cemetery; I saw the rotted dust and bones that had once deliciously been Beatriz Viterbo; I saw the circulation of my own dark blood; I saw the coupling of love and the modification of death; I saw the Aleph from every point and angle, and in the Aleph I saw the earth and in the earth the Aleph and in the Aleph the earth; I saw my own face and my own bowels; I saw your face; and I felt dizzy and wept, for my eyes had seen that secret and conjectured object whose name is common to all men but which no man has looked upon — the unimaginable universe.

-Jorge Luis Borges, “The Aleph”

Aleph (\aleph) is of course the letter chosen by Georg Cantor to represent transfinite cardinals and the first letter of the Hebrew alphabet. It plays a special role in Kabbalah as the first letter in “Ein Sof,” roughly translated as “infinity,” and in “Elohim,” one of the names of the Hebrew god. We will surely return to these matters.

This is the first installment in a mini-series on what we will call “universal structures,” objects that contain all other objects of their type. We will continue to look at examples from literature and religion, and will delve into the existence of universal structures in mathematics, a topic which continues to drive cutting-edge research to this day. Next week, we will look at a particular universal structure in mathematics, the wonderfully named “random graph.” I hope you will join us.

An Infinitude of Proofs, Part 1

In our previous post, we gave three elementary number-theoretic proofs of the infinitude of the prime numbers. Today, in an unforgivably delayed second installment, we provide two proofs using machinery from more distant fields of mathematics: analysis and topology.

A small word of warning: today’s proofs, though not difficult, require a bit more mathematical sophistication than those of the previous post. The first proof will involve some manipulation of infinite sums, and the second proof uses rudiments of topology, which we introduced here.

Without further ado, let us begin. Our first proof today dates to the 18th century and is due to the prolific Leonhard Euler, of whom Pierre-Simon Laplace, an exceptional mathematician in his own right, once said, “Read Euler, read Euler, he is the master of us all.”

Proof Four (Euler): Suppose there are only finitely many prime numbers, \{p_1, p_2, \ldots, p_n\}. For each prime number p, consider the infinite series \sum_{i = 0}^\infty \frac{1}{p^i} = 1 + \frac{1}{p} + \frac{1}{p^2} + \ldots . As you probably learned in a precalculus class, the value of this infinite sum is precisely \frac{1}{1-\frac{1}{p}}.

Now recall that every natural number greater than 1 can be written in a unique way as a product of prime numbers. Since \{p_1, \ldots, p_n\} are all of the primes, this means that, for each natural number m \geq 1, we can express \frac{1}{m} as \frac{1}{p_1^{s_1}} \cdot \frac{1}{p_2^{s_2}} \cdot \ldots \cdot \frac{1}{p_n^{s_n}} for some natural numbers s_1, s_2, \ldots, s_n. Moreover, it is clear that this gives a one-to-one correspondence between natural numbers m \geq 1 and the corresponding n-tuples of natural numbers, \langle s_1, s_2, \ldots, s_n \rangle. Putting this all together, we obtain the following remarkable formula:

\sum_{m=1}^\infty \frac{1}{m} = \prod_{k=1}^n \frac{1}{1-\frac{1}{p_k}}.

The sum on the left hand of this equation is the famous harmonic series, and it is well-known and easily verified that this sum diverges to infinity. On the other hand, the product on the right hand is a finite product of finite numbers and is therefore finite! This gives a contradiction and concludes the proof.

Our next proof was published in 1955 by a young Hillel Furstenburg, who is still active at the Einstein Institute of Mathematics here in Jerusalem.

Proof Five (Furstenberg): For all integers a and b, let U_{a,b} be the set \{a + bk \mid k \in \mathbb{Z} \}, i.e., U_{a,b} contains a and all integers that differ from a by a multiple of b.

Claim: For all pairs of integers (a_0, b_0) and (a_1, b_1), either U_{a_0,b_0} \cap U_{a_1, b_1} = \emptyset or there are a_2, b_2 such that U_{a_0,b_0} \cap U_{a_1, b_1} = U_{a_2, b_2}.

Proof of Claim: If U_{a_0, b_0} \cap U_{a_1, b_1} \neq \emptyset, then let a_2 be any element of the intersection, and let b_2 be the least common multiple of b_0 and b_1. It is easily verified that U_{a_0,b_0} \cap U_{a_1, b_1} = U_{a_2, b_2}, thus finishing the proof of the claim.

It now follows that we can define a topology on \mathbb{Z} by declaring that a set U \subseteq \mathbb{Z} is open if and only if U = \emptyset or U is a union of sets of the form U_{a,b}. In particular, for each prime number p, the set U_{0,p}, which is the set of all multiples of p, is open. However, U_{0,p} is also closed, since it is the complement of the set \bigcup_{0 < a < p} U_{a,p}, which is an open set.

Now suppose that there are only finitely many prime numbers, \{p_1, \ldots, p_n\}. Since each set U_{0,p_k} is closed and a finite union of closed sets is closed, we have that \bigcup_{k = 1}^n U_{0,p_k} is a closed set. Notice that every integer that is not 1 or -1 is a multiple of some prime number, so \bigcup_{k = 1}^n U_{0,p_k} is precisely the set of all integers except 1 and -1. Since it is closed, its complement, which is \{-1,1\}, is open. But every non-empty open set is a union of sets of the form U_{a,b}, each of which is clearly infinite, so there can be no finite non-empty open sets. This is a contradiction and concludes the proof.

Cover Image: Passio Musicae by Eila Hiltunen, a monument to Jean Sibelius in Helsinki, Finland. Photograph by the author.