My book *Notes on Counting: An Introduction to Enumerative Combinatorics* should be published by Cambridge University Press in June this year, as part of the Australian Mathematical Society Lecture Series.

If you have read some of my lecture notes on enumerative combinatorics, you may want to know that this book contains more than just the union of all the notes I have written on this subject!

If you are interested, there is a flyer, and preorder form, here.

But no job is ever finished. In the book, at the start of Chapter 4, I prove (as an introductory example for recurrence relations) that the number of compositions of *n* (ordered sequences of positive integers with sum *n*) is 2^{n−1} if *n* ≥ 1. The easy proof involves showing that the number of compositions of *n* is twice the number of compositions of *n*−1. When I came to mark my St Andrews exam, I found that two students had produced a beautifully short and elegant proof of this, which may not be new but was certainly new to me!

It goes like this. Imagine you are a typesetter; you set up a row of 2*n*−1 boxes, each to contain a symbol. In the odd numbered boxes you put a 1; in the even numbered boxes, you choose to put a plus sign or a comma. Your 2^{n−1} choices give all the compositions of *n*, each once.

For example, the compositions of 3 are

1+1+1 1+1,1 1,1+1 1,1,1

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What follows are random musings on what we heard in a very enjoyable two days.

Three speakers mentioned Galton–Watson branching processes in similar contexts: **Andrew Treglown**, **Oliver Riordan** and **Guillem Perarnau**. I got two insights from this, which I had probably not realised clearly before. First, Erdős–Rényi random graph is the same thing as percolation on the complete graph. (For percolation on a graph, we allow edges to be open independently with probability *p* and ask questions about whether the resulting graph has a “giant component” and similar things. Erdős and Rényi simply included edges with probability *p* from all 2-subsets of the vertex set, and asked similar questions.)

Secondly, for percolation in any regular graph of valency *d*, to explore the component containing *v* you look at the neighbours of *v*, their neighbours, and so on until you have seen everything. The number of neighbours of any vertex (apart from the vertex on the path from *v* by which we reached it) is a binomial random variable Bin(*d*−1,*p*), and the whole process is “stochastically dominated” by the Galton–Watson process with this rule for descendents. (In an infinite tree, they would be identical; but the possibility of short cycles means that we will not see more vertices in the percolation case than in the Galton–Watson case.)

The three speakers tackled different problems. Perarnau was looing through the critical window in the percolation problem for regular graphs. Treglown was examining resilience of random regular graphs, e.g. how many edges can you delete before losing the typical value of a Ramsey number in the graph. (As he said, there are two processes going on here: first choose a random graph, then delete an arbitrary subset of its edges.) Riordan was considering the Achlioptas process, a variant on Erdős–Rényi where, at each step, you choose two random edges, and keep one, the choice depending on the size of the components they connect.

**Kitty Meeks** talked about complexity problems for sum-free sets. How do such problems arise? She began with a theorem of Erdős, according to which any set of *n* natural numbers contains a sum-free subset of size at least *n*/3. Now one can ask whether there is a sum-free subset of size *n*/3+*k*, or to count such subsets. This kind of problem lends itself very well to analysis via *parameterised complexity*.

**Sophie Huczynska** talked about the “homomorphic image” order on relational structures such as graphs: in this order, *G* is smaller than *H* if there is a homomorphism from *G* to *H* mapping the edge set of *G* onto the edge set of *H*. (She began by describing a variety of graph orders, such as subgraph order, induced subgraph order, and minor order, and showing how all can be expressed in terms of homomorphisms.) Typical questions are whether a class of graphs or other structures is a *partial well-order* with respect to the homomorphic image order (that is, whether it contains no infinite antichains), and describing the antichains if possible.

A special mention for **Ewan Davies**, who stepped in at very short notice when Agelos Georgakopoulos was unable to speak. (We found out the reason later; his wife had produced their first son that very day.) Evan gave a fascinating talk on getting accurate values or estimates for the coefficients of the matching polynomial of a graph (the polynomial in which the coefficient of *x ^{k}* is the number of matchings with

I am afraid I didn’t feel up to having drinks afterwards, so just sneaked off home.

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I don’t intend to give a blow-by-blow account of all this.

The big theorem says that there is an algorithm which tests isomorphism of two *n*-vertex graphs in time bounded by exp(*O*(log *n*)^{c}) for some constant *c*. As the name might suggest, the bound is polynomial if the constant *c* is equal to 1. The best previous result for graph isomorphism was exp(*O*(*n*^{c})), that is, fractional exponential.

The strategy is “divide and conquer”, and as Laci explained, his job was to divide, since the conquering had already been achieved by Luks. The dividing is rather involved, and I won’t attempt to describe it. Any graph isomorphism algorithm has to name an object in the graph; this incurs a multiplicative cost equal to the number of objects which could have been named.

The basic division step is called “split or Johnson”. It identifies, at a cost which can be controlled, either a small subset of the vertex set, or a partition of the vertex set, or a *Johnson graph* (whose vertices are the *k*-element subsets of an auxiliary *m*-element set). But for the induction, we have to have the structure of a graph on the auxiliary set, and further complicated reductions were needed to achieve this.

In the second tutorial, he described carefully the mistake that Harald Helfgott had found in the proof, and how he had fixed it. The arguments seemed fairly familiar to me; there are similar things in my Oxford DPhil thesis in 1971. This was the time when the graph-theoretic methods introduced into permutation group theory by Charles Sims and Donald Higman were beginning to make their mark on permutation group theory.

Nothing is ever lost!

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Nor is it about important factors, highlighted in the 2011 International Review of UK Mathematics although largely ignored by the research council that commissioned it: research area, size of research group, and size of institution.

But I discovered recently a diversity in the way we do mathematics, which I found surprising and potentially significant.

A few weeks ago, I was at dinner with a visiting colloquium speaker. The conversation turned to whether mathematical thought is done in words, or is “pre-linguistic”.

This is a topic about which Jacques Hadamard, in his book originally called *The Psychology of Invention in the Mathematical Field* but re-published as *The Mathematician’s Mind*, had a lot to say. Some linguists and linguistic philosophers, notably Max Müller, insist that language is essential to thought, and that no thoughts can be pre-linguistic. Hadamard, from his own intuition and from the writings of others from Poincaré to Einstein, is convinved that this is not the case, and is bewildered that Müller can hold this view with such vehemence. In a footnote, he says,

I have also seen the following topic (a deplorable subject, as far as I can judge) proposed for an examination—an elementary one, the “baccalauréat”—in philosophy in Paris: “To show that language is as necessary for us to think as it is to communicate our thoughts.”

For me, I know for sure that my best insights (those which are not just routine calculations) are pre-linguistic, and I struggle to put them into words: similarly, if the insight is a conjecture, I struggle to see how the conjecture might be proved. I assumed that most mathematicians would be like me, and would agree with Hadamard rather than Müller.

So it was a bit of a surprise when, of the five research mathematicians at the table, we were split 3 to 2 in Hadamard’s favour.

This is of course an anecdote, and not survey data. But we noticed a curious thing. The two who said they did mathematics in words had something probably significant in common: their cradle tongue (in both cases, a Slavic language) was not the language in which they do mathematics (in both cases, English); and both of them had learnt English at a comparatively advanced age. The other three of us were all native English speakers.

Not sure what to make of this. But I am glad that it drove me back to Hadamard’s book. I had completely forgotten that, at a certain point, he admits to his failure to be able to think creatively about group theory!

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The *power graph* of a group *G* has vertex set *G*, with an edge from *x* to *y* if one is a power of the other; the *directed power graph* is a directed graph with an arc from *x* to *y* if *y* is a power of *x*. Various studies have looked at the properties of these graphs as graphs. But the questions discussed here are:

As you would expect, all these questions have negative answers in general.

For finite groups, it is known that the power graph determines the directed power graph up to isomorphism. But it does not determine it uniquely. (In the cyclic group of order 6, the identity and the two generators are all joined to everything, and so the power graph does not determine which element is the identity; the directed power graph clearly does.) The power graph does not determine the group up to isomorphism: for example, any group of exponent 3 has power graph consisting of a number of triangles with a common vertex.

For infinite groups, things are worse. The Prüfer group **Z**_{p∞} (the group of rationals with *p*-power denominators modulo the group of integers, where *p* is prime) has the property that its power graph is complete, so we cannot even determine the prime, whereas clearly we can determine *p* from the directed power graph.

It seems to be the fact that all elements have finite order that causes the trouble here. So let us banish them, and assume that the group is torsion-free. These groups have the great advantage that, if *x* = *y ^{n}*, then

At once a small problem of definition arises. The identity element is equal to *x*^{0} for any element *x*, so is joined to everything in the power graph. In particular, in the infinite cyclic group, we cannot distinguish the identity from the two generators (much as happens for the cyclic group of order 6 in our earlier example). But, in any other torsion-free group, the identity is the only vertex joined to everything. So, if we change the definition slightly so that edges or arcs are not put in from *x* to *x*^{0}, then the identity becomes isolated in any torsion-free group (and only in such groups), and apart from the infinite cyclic group the answers to our questions will not be changed. So we simplify things slightly by making this change.

Now here are some of the results.

- If the power graph of
*H*is isomorphic to that of the infinite cyclic group**Z**, then*H*is isomorphic to**Z**, and any power graph isomorphism is a directed power graph isomorphism. [It need not be a group isomorphism, since an element and its inverse have the same neighbours, and so may be interchanged by a graph isomorphism.] - If two torsion-free nilpotent groups of class (at most) 2 have isomorphic power graphs, then they have isomorphic directed power graphs.
- If
*G*is a torsion-free group in which any non-identity element lies in a unique maximal cyclic subgroup (free and free abelian groups are examples of this), then the power graph of*G*is a disjoint union of connected components, each isomorphic to the power graph of**Z**with the identity removed, together with one isolated vertex. In particular, if two groups with these properties have the same cardinality, and neither is**Z**, then they have isomorphic power graphs; and any power graph isomorphism is a directed power graph isomorphism.

There are also some results about **Q**, the additive group of rationals, and some of its subgroups. For **Q** we have the interesting property that any power graph isomorphism is either an isomorphism or an anti-isomorphism (reversing all directions) of the directed power graph.

The main authors of this paper are St Andrews undergraduates Horacio Guerra and Šimon Jurina, who proved these results as part of their summer research project last year. On Wednesday they took time out from exam revision to present these results in our Algebra and Combinatorics seminar. I wish them good fortune in their exams and in their subsequent careers.

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I described this here, and went on to explain how Sebi Cioaba and I showed that for any *m* ≥ 2 it is possible to cover the edges of the complete graph *m* times with 3*m* Petersens. In particular, you can cover the edges twice with six Petersen graphs.

This means that five Petersen graphs suffice to cover the edges of the complete graph, since we may omit one of the six. We cannot, however, omit two, since any two have edges in common.

So the minimum number of Petersen graphs required to cover all the edges is either 4 or 5.

In fact it is 4. If you apply four random permutations to the vertices of the Petersen graph you will quickly come up with a way of covering all edges.

But is there a nice way to see this? In particular, is it possible to arrange four Petersen graphs so that they cover each edge once or twice (so that deleting a complete graph leaves a trivalent simple graph remaining)? If so, what is this graph? If not, why not?

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The book is structured as a sequence of interviews with celebrated manuscripts, as if they were human stars; we meet them in the libraries where they now reside, and are given an impression of their outward appearance before investigating the contents, history, and unexpected connections. The manuscripts include the Book of Kells and an early manuscript of the *Carmina Burana*.

Reading it, I got quite a surprise. But let me fill in some background first.

The pre-Socratic Greek philosophers where highly inventive and speculative. As well as arguing about whether “everything is change” or “change is impossible”, or whether the world is made out of water or fire or something else, they developed a number of models of the universe. Among these were a geocentric model rather like the one that prevailed until the time of Copernicus, and a heliocentric model (similar to that proposed by Copernicus) put forward by Aristarchus; and also two intermediate models.

One of these, the “Egyptian” system of Herakleides, took into account the fact that Mercury and Venus behave very differently from the other planets, both in their sticking relatively close to the Sun rather than wandering around the whole Zodiacal belt, and in their waxing and waning in brightness as they moved. In hindsight it seems clear that, however the outer planets behave, there is clear evidence that the inner planets go round the Sun. The model proposed by Herakleides involved Mercury and Venus circling the Sun as it (like the other planets) moved round the earth.

A more drastic revision had all the other planets (except the moon) circling the Sun as it moved round the earth: this model was later and perhaps independently proposed by Tycho de Brahe as a way of getting some of the advantages of the Copernican system while not falling foul of those who insist that the earth is fixed.

Then along came Plato, who said that the most perfect form of motion is uniform motion in a circle; and Aristotle, who said that change and decay exist only in the sublunary sphere, and so all the planets must undergo uniform circular motion. This view held sway for more than a millennium and a half.

Or at least, so goes received wisdom. But one of de Hamel’s manuscripts is the *Aratea*, in the university library in Leiden. It deals with astronomy, with vivid pictures of the constellations, but also includes a “planetarium”, which clearly shows Herakleides’ model with Mercury and Venus circling the Sun.

This manuscript was produced at the court of Charlemagne, in his lifetime or a little after. It is a copy of a manuscript explaining the theories of Aratus of Soli, an astronomer who lived about 300BC, rendered into Latin. The original has not been found; according to de Hamel, a new copy was more important at the time than a possibly battered and damaged original, which could be thrown away once the copy had been made! But this shows clearly that some pre-Socratic Greek knowledge had not been lost by around the year 800. In fact, the planetarium is not part of Aratus’ work, and is probably taken from a calendar from the year 354.

Even more intriguingly, de Hamel cites the work of modern astronomers who have examined this planetarium closely. If we assume that the scribes recorded the actual configuration of planets and stars at the time, and worked to 15 degrees accuracy, the configuration shown occurred on 18 March and 14 April 816, and not again (or before) for 98000 years.

If this is correct, it gives us an astonishingly accurate dating of a mediaeval manuscript. According to a calculation by Bede a little earlier, 18 March was the date on which God created the universe (in the year 3952BCE), so this would be an important anniversary. Charlemagne had died two years earlier, and his son Louis the Pious would be crowned later in the year 816, so the date is plausible historically. There were dramatic developments in education and study in the reign of Charlemagne, driven by Alcuin of York who was his adviser in this early renaissance of classical learning.

Arthur Koestler, from whose book *The Sleepwalkers* I have taken the account of Greek cosmology, admits that by the year 1000 some of the alternative Greek cosmological manuscripts were being rediscovered; but it seems that this happened earlier, or perhaps in some sense they had never been forgotten. Clearly there are currents here that I am not aware of!

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Any real number *x* in [0,1] has a base 2 expansion. The density of 1s in this expansion (the limit of the ratio of the number of 1s in the first *n* digits to *n*) need not exist; but, according to the Strong Law of Large Numbers, for almost all *x*, the limit exists, and is 1/2.

So it might seem perverse to consider the set *X _{p}* of numbers for which the limit is equal to

Briefly, given a subset of [0,1], take a cover of it by intervals of length at most δ, and take the sum of the *s*th powers of the lengths of the intervals. Now take the infimum of this quantity over all such coverings, and the limit of the result as δ tends to 0. The result is the *s*-dimensional Hausdorff measure of the set. There is a number *s*_{0} such that the measure is ∞ for *s* < *s*_{0}, and is 0 for *s* > *s*_{0}; for *s* = *s*_{0}, it may take any value. This *s*_{0} is the *Hausdorff dimension* of the set.

Besicovich calculated the Hausdorff dimension of the sets *X _{p}* defined earlier. It turns out to be equal to the

The entropy function suggests a relation to binomial coefficients and Stirling’s formula, which is indeed involved in the proof. (The logarithm of the binomial coefficient {*n* choose *pn*} is asympototically *nH*(*p*), as follows easily from Stirling’s approximation for *n*!.)

All this can be phrased in terms of the dynamics of the map 2*x* mod 1 on the unit interval (which acts as the left shift on the base 2 expansion), which suggests a good direction for generalisation, and suggests too that that this generalisation will involve concepts from ergodic theory such as entropy and pressure. Most of the lecture was about this.

(Perhaps worth mentioning that all these sets are negligible in the sense of Baire category, according to which almost all real numbers have the property that the lim inf of the density is 0 and the lim sup is 1.)

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At the weekend we had two beautiful spring days; bluebells are out in force, and rarer plants such as water avens were putting on good displays. Monday morning dawned sunny, but by lunchtime the weather had reverted briefly to winter. Since everything was in the Mathematical Institute this didn’t matter too much.

Here are notes on just a few of the talks that struck me.

The opening lecture was by **Simon Blackburn**, who talked about network coding (a subject I knew a bit about, thanks to working with Søren Riis and Max Gadouleau at Queen Mary; I wrote about it here. Briefly, in a network through which some physical commodity flows, there can be bottlenecks; but if the network carries data, then giving the nodes some simple processing power can allow faster transmission. The famous butterfly network illustrates this; I started my account with this, and Simon also began with it.

There are *n* packets of information which have to be sent through the network to a number of sinks. Unlike the simple butterfly, real networks are large and complicated, and so a sink will not know which functions of the input packets it is receiving, and so will not know how to recover the original messages. If (as we assume) the operations of intermediate nodes are linear, then all that the sinks know is that what they receive lies in the subspace of *F ^{n}* spanned by the input data. So we have to shift our viewpoint, and regard the information being sent as a subspace of a vector space, rather than a specific collection of vectors.

The number of *k*-dimensional subspaces of an *n*-dimensional vector space over a field of order *q* is a *Gaussian* or *q-binomial coefficient*. As my Advanced Combinatorics students know, this is a monic polynomial in *q* of degree *k*(*n*−*k*). So, for large *q*, it is about *q*^{k(n−k)}. This number is equal to the number of subspaces which are the row spaces of matrices of the form (*I*, *D*) where *I* is the identity matrix of order *k* and *D* an arbitrary *k*×(*n*−*k*) matrix. So we can restrict attention to these. Whatever the sink receives is a matrix with the same row space; by performing row operations it can reduce it to the above form and recover the data vector *D*.

What if some nodes in the network are not working correctly? This could be either because of mechanical faults or the result of malicious hacking. If *t* nodes are affected, then the actual subspace received at the sink will be at distance *t* or less from the correct one, where the distance between two subspaces *U* and *W* is the maximum of dim(*U*)−dim(*U*∩*W*) and dim(*W*)−dim(*U*∩*W*). So we have a new setting in which to do error-correction, with subspaces replacing words and the above distance replacing Hamming distance. Simon explained the analogue of the Johnson bound for constant-dimension subspace codes, and some recent work of his with Tuvi Etzion and Jessica Claridge on this.

**Kitty Meeks** talked about a new version of graph colouring, *interactive sum list colouring*. In regular graph colouring we have a fixed set of *k* colours, and ask for the smallest *k* for which a given graph can be coloured. List colouring generalises this, by giving each vertex a list of *k* colours from a possibly much larger set of colours and again asking for the smallest *k* for which the graph can be coloured. Sum list colouring extends further, by allowing the lists given to vertices to be of different sizes, and minimizing the sum of the list sizes. The new generalisation takes this one step further, and is best thought of as a game between Alice, who is trying to colour a graph, and Bob, who is trying to thwart her. At each round Alice is allowed to ask for a new colour for a particular vertex, different from those already provided by Bob for that vertex; each request has a cost of 1, and she is trying to minimise (and Bob to maximise) the total cost. The corresponding colouring number is the cost if both Alice and Bob play optimally.

Let scn(*G*) and iscn(*G*) be the minimum sums required in the two cases. An easy argument shows that scn(*G*) is at most the sum of the numbers of vertices and edges of *G*; we call *G* *sc-greedy* if this bound is met. Many examples of sc-greedy graphs are known.

Not too much is known yet about iscn(*G*). It is an interesting exercise to show that the three-vertex path has iscn equal to 4. (Ask for a colour for each vertex first, and then consider possible cases.) This graph has scn equal to 5 (it is sc-greedy).

It is known that iscn(*G*) does not exceed scn(*G*), and is equal to it in the case of complete graphs; they (Kitty and her collaborator Marthe Bonamy) have shown that it is strictly smaller for other graphs. Indeed, the difference between the two numbers is at least (*n*−ω(*G*))/2, where ω(*G*) is the clique number of *G*.

The first day concluded with a talk by **Max Gadouleau**. As I said before, Max has done some good work on network coding, but in this talk he stepped back. Networks are studied in many areas of mathematics and science, and there is inevitably a certain amount of multiple discovery and of calling the same thing by distinct names.

Max actually talked about *discrete dynamical systems*. A dynamical system is just a set with a function on it, and we are interested in iterating the function. If the set has *N* elements, there are *N ^{N}* functions, and the long-term behaviour is simple: there is a subset of the points on which the function acts as a permutation (the

The connection with networks arises thus. We are particularly interested in the case where there is a set *V* of *nodes*, each of which can be in one of a finite number *q* of *states*; so we have *N* = *q ^{n}*, a function

Apparently this has real meaning. Fixed points in a gene regulation system may, for example, correspond to different modes of functioning of the cell, and may include cancer.

One thing that one can do now is to ask about dynamical systems with a given interaction graph. Another is to fix the graph and vary *q*, or restrict the kind of functions allowed (linear, monotone, threshold, etc.). Yet again we could have different rules for the updates. Max considered only the case where all vertices update simultaneously.

One of the few things known is the analogue of the Singleton bound in coding theory: the number of fixed points is at most *q*^{τ}, where τ is the size of the smallest *feedback set* (a set whose complement contains no directed cycles). Max gave exact values for the size of the image and the number of periodic points in terms of graph-theoretic parameters of the interaction graphi, such as the maximum number of independent arcs, or the maximum number of vertices covered by disjoint cycles. (One of these equalities holds only for *q* > 2, and is just a bound in the case *q* = 2.)

The next day, **David Bevan** gave the first of two nice talks on permutation patterns. (I have discussed these here also.) Briefly, a short permutation $pi; is involved in a longer permutation σ if, when we remove some points and push everything down to the shorter interval, we obtain π. (In my preferred way of thinking about it, a permutation is a pair of total orders on a set; involvement is just the “induced substructure” relation.) A permutation of length *n* is *k-prolific* if every permutation of length *n*−*k* is involved in it. They (David and co-authors Cheyne Homberger and Bridget Tenner) have found that *k*-prolific permutations of length *n* exist if and only if *n* ≥ *k*^{2}/2+2*k*+1. He outlined the proof, which involves an interesting detour through packings of diamond-shaped tiles in the plane.

**Anders Claesson** gave us a number of different proofs that there the same number of subsets of even and odd size of an *n*-set for non-zero *n*. The proofs, of increasing complexity, involved manipulations of exponential generating functions, and in particular the use of sign-reversing involutions. He went on to further applications, including an explicit count for interval orders. He ended up by remarking that species form a semiring, which can be extended to a ring of *virtual species* in the same way that the natural numbers are extended to the integers, and that these give a way of handling “negative objects”.

**Robert Brignall** talked about the notorious problem of counting permutations excluding 1324. He quoted Doron Zeilberger: “Even God doesn’t know the value of Av(1324)_{1000}“. Not even the exponential constant is known, although simulations suggest it is around 11.6. Robert and his co-authors (David Bevan, Andrew Elvey Price and Jay Pantone) have improved the upper and lower bounds for this constant to 13.5 and 10.24, both improvements on previously known bounds. The ingenious proof encloses the diagram of such a permutation in a staircase made up of dominoes with similar structure, but is difficult to describe without pictures!

The conference closed with a talk by **Laszlo Babai**. Laci is in St Andrews for a week, and will be giving five talks, including two ninety-minute talks at the British Colloquium on Theoretical Computer Science today and tomorrow. So I will defer my comments on his talks for a while …

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My second talk was on sum-free sets, and was rather more discursive, so here is a summary.

A set of natural numbers is *k-AP-free* if it contains no *k*-term arithmetic progression, and is *sum-free* if it contains no solution to *x*+*y* = *z*.

Three big theorems in additive combinatorics are:

**Theorem 1 (van der Waerden)** For a natural number *k* > 2, the set **N** cannot be partitioned into finitely many *k*-AP-free sets.

**Theorem 2 (Roth–Szemerédi** For a natural number *k* > 2, a *k*-AP-free set has density zero.

**Theorem 3 (Schur)** The set **N** cannot be partitioned into finitely many sum-free sets.

At first sight, one would like a theorem to complete the pattern, asserting that a sum-free set has density zero. But this is false, since the set of all odd numbers is sum-free and has density 1/2. What follows could be motivated as an attempt to find a replacement for the missing fourth theorem.

The *Cantor space* *C* can be represented as the set of all (countable) sequences of zeros and ones. It carries the structure of a complete metric space (the distance between two sequences is a monotonic decreasing function of the index of the first position where they differ) or as a probability space (corresponding to a countable sequence of independent tosses of a fair coin).

We define a bijection between Cantor space and the set **S** of all sum-free subsets of **N**. Given a sequence *x* in *C*, we construct *S* as follows:

Consider the natural numbers in turn. When considering *n*, if *n* is the sum of two elements already put in *S*, then of course *n* is not in *S*. Otherwise, look at the first unused element of *x*; if it is 1, then put *n* into *S*, otherwise, leave *n* out of *S*. Delete this element of the sequence and continue.

For example, suppose that *x* = 10110…

- The first element of
*x*is 1, so 1 ∈*S*. - 2=1+1, so 2 is not in
*S.* - 3 is not in
*S*+*S*; the next element of*x*is 0, so 3 is not in*S*. - 4 is not in
*S*+*S*; the next element of*x*is 1, so 4 is in*S*. - 5=1+4, so 5 is not in
*S*. - 6 is not in
*S*+*S*; the next element of*x*is 1, so 6 is in*S*. - …

So *S* = {1,4,6,…}.

The notion of “almost all” in a complete metric space is a *residual set*; a set is residual if it contains a countable intersection of dense open sets. Thus, residual sets are non-empty (by the Baire Category Theorem); any countable collection of residual sets has non-empty intersection; a residual set meets every non-empty open set; and so on.

A sum-free set is called *sf-universal* if everything which is not forbidden actually occurs. Precisely, *S* is sf-universal if, for every subset *A* of {1,…,*n*}, one of the following occurs:

- there are
*i,j*∈*A*with*i*<*j*and*j−i*∈*S*; - there exists
*N*such that*S*∩[*N*+1,…,*N*+*n*] =*N*+*A*,

where *N*+*A* = {*N*+*a*:*a*∈*A*}.

**Theorem** The set of sf-universal sets is residual in **S**.

**Theorem (Schoen)** A sf-universal set has density zero.

Thus our “missing fourth theorem” asserts that almost all sum-free sets (in the sense of Baire category) have density zero.

There is a nice application. Let *S* be an arbitrary subset of **N**. We define the *Cayley graph* Cay(**Z**,*S*) to have vertex set **Z**, with *x* joined to *y* if and only if |*y−x*|∈*S*. Note that this graph admits the group **Z** acting as a shift automorphism on the vertex set.

**Theorem**

- Cay(
**Z**,*S*) is triangle-free if and only if*S*is sum-free. - Cay(
**Z**,*S*) is isomorphic to Henson’s universal homogeneous triangle-free graph if and only if*S*is sf-universal.

So Henson’s graph has uncountably many non-conjugate shift automorphisms.

In a probability space, large sets are those which have measure 1, that is, complements of null sets. Just as for Baire category, these have the properties one would expect: the intersection of countably many sets of measure 1 has measure 1; a set of measure 1 intersects every set of positive measure; and so on.

The first surprise is that measure and category give entirely different answers to what a typical set looks like:

**Conjecture** The set of sf-universal sets has measure zero.

Although this is not proved yet (to my knowledge), it is certain that this set does not have measure 1.

Given the measure on **S**, and our interest in density, it is natural to ask about the density of a random sum-free set. This can be investigated empirically by computing many large sum-free sets and plotting their density. Here is the rather surprising result.

The spike on the right corresponds to density 1/4 and is explained by the following theorem.

**Theorem**

- The probability that a random sum-free set consists entirely of odd numbers is about 0.218 (in particular is non-zero).
- Conditioned on this, the density of a random sum-free set is almost surely 1/4.

A word about this theorem. Suppose that we have tossed the coin many times and found no even numbers. Then we have chosen the odd numbers approximately independently, so the next even number is very likely to be excluded as the sum of two odd numbers, whereas the next odd number is still completely free. So the probability of no even numbers is not much less than the probability of no even numbers in the first *N* coin tosses. Moreover, since the odd numbers are approximately independent, we expect to see about half of them, and so about a quarter of all numbers.

Other pieces can also be identified. Let **Z**/*n***Z** denote the integers modulo *n*. We can define the notion of a sum-free set in **Z**/*n***Z** in the obvious way. Such a sum-free set *T* is said to be *complete* if, for every *z* in (**Z**/*n***Z**)\*T*, there exist *x,y* in *T* such that *x*+*y* = *z* in **Z**/*n***Z** . Now the theorem above extends as follows. Let **S**(*n,T*)$ denote the set of all sum-free sets which are contained in the union of the congruence classes *t* (mod *n*) for *t*∈*T*.

**Theorem** Let *T* be a sum-free set in **Z**/*n***Z**.

- The probability of
**S**(*n,T*) is non-zero if and only if*T*is complete. - If
*T*is complete then, conditioned on*S*∈**S**(*n,T*), the density of*S*is almost surely |*T*|/2*n*.

In the figure it is possible to see “spectral lines” corresponding to the

complete modular sum-free sets {2,3} (mod 5) and {1,4} (mod 5) (at density 1/5) and {3,4,5} (mod 8) and {1,4,7} (mod 8) (at density 3/16).

The density spectrum appears to be discrete above 1/6, and there is some

evidence that this is so. However, a recent paper of Haviv and Levy shows

the following result.

**Theorem** The values of |*T*|/2*n* for complete sum-free sets *T* in **Z**/*n***Z** are dense in [0,1/6].

However, this is not the end of the story. Neil Calkin and I showed that the event that 2 is the only even number in *S* has positive (though rather small) probability. More generally,

**Theorem** Let *A* be a finite set and *T* a complete sum-free set modulo *n*. Then the event *A* ⊆ *S* ⊆ *A*∪(*T* (mod *n*)) has positive probability.

**Question** Is it true that a random sum-free set almost surely has a density? Is it further true that the density spectrum is discrete above 1/6 but has a continuous part below 1/6? Is it also true that the probability that a sum-free set has zero density is 0?

In this connection, consider the following construction of Calkin and Erdős. Let α be an irrational number, and define *S*(α) to be the set of natural numbers *n* for which the fractional part of *n*α lies between 1/3 and 2/3. It is easy to see that *S*(α) is sum-free and has density 1/3. However this does not resolve the question, since the event *S* ⊆ *S*(α) for some α has probability zero.

However, there might be other examples along these lines …

A sequence *x* is *ultimately periodic* if there exist positive integers *n* and *k* such that *x*_{i+k} = *x _{i}* for all

It is easy to see that, in our bijection from sequences to sum-free sets, a sequence which maps to an ultimately periodic sum-free set must itself be ultimately periodic. What about the converse?

**Question** Is it true that the image of an ultimately periodic sequence is an ultimately periodic sum-free set?

After some investigation, Neil Calkin and I conjectured that the answer is “no”. There are some ultimately periodic sequences (the simplest being 01010 repeated) for which no sign of periodicity has been detected despite computing nearly a million terms. These sets are fascinating, and seem sometimes to exhibit an “almost periodic” structure; they settle into a period, which is then broken and a longer period established, and so on.

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