Yufei Zhao

Category: Paper

A reverse Sidorenko inequality

Together with undergraduates Ashwin Sah, Mehtaab Sawhney, and David Stoner (the same team that proved Kahn’s conjecture on independent sets that I blogged about earlier), we are excited to announce our new paper A reverse Sidorenko inequality. This paper solves several open problems concerning graph colorings and homomorphisms, including one of my favorite problems regarding maximizing the number of q-colorings in a d-regular graph. I had previously blogged about some of these problems.

It has been truly a pleasure working with Ashwin, Mehtaab, and David on this project. I have learned so much from them.


Read the rest of this entry »


The number of independent sets in an irregular graph

I am excited to announce our new paper, The number of independent sets in an irregular graph, coauthored with these three amazing collaborators:

Ashwin is a freshman at MIT, Mehtaab is a sophomore at MIT, and David is a junior at Harvard.

The paper solves a conjecture made by Jeff Kahn in 2001 concerning the number of independent sets in a graph.

An independent set of a graph is a subset of vertices with no two adjacent. For example, here is the complete list of independent sets of a cycle of length 4:


Since many problems in combinatorics can be naturally phrased in terms of independent sets in a graph or hypergraph, getting good bounds on the number of independent sets is a problem of central importance.

Instead of giving the exact statement of the conjecture (now theorem), which you can find in the abstract of our paper, let me highlight a specific instance of a problem addressed by the theorem:

Consider the family of graphs with no isolated vertices and

  • exactly a third of the edges have degree 3 on both endpoints, and
  • a third of the edges have degree 4 on both endpoints,
  • the remaining third of the edges have degree 3 on one endpoint and degree 4 on the other endpoint.

What is the smallest constant {c} such that every {n}-vertex graph satisfying the above properties has at most {c^n} independent sets?

In other words, letting {i(G)} denote the number of independent sets and {v(G)} the number of vertices of {G}, maximize the quantity {i(G)^{1/v(G)}} over all graphs {G} in the above family.

(See the end of this post for the answer)

In summer 2009, as an undergraduate participating in the wonderful Research Experience for Undergraduates (REU) run by Joe Gallian in Duluth, MN, I learned about Kahn’s conjecture (somewhat accidentally actually, as I was working on a different problem that eventually needed some bounds on the number of independent sets, and so I looked up the literature and learned, slightly frustratingly at the time, that it was an open problem). It was on this problem that I had written one of my very first math research papers.

I had been thinking about this problem on and off since then. A couple years ago, Joe Gallian invited me to write an article for a special issue of the American Mathematical Monthly dedicated to undergraduate research (see my previous blog post on this article), where I described old and new developments on the problem and collected a long list of open problems and conjectures on the topic (one of them being Kahn’s conjecture).

This spring, I had the privilege with working with Ashwin, Mehtaab, and David, three energetic and fearless undergraduate students, finally turning Kahn’s conjecture into a theorem. Fearless indeed, as our proof ended up involving quite a formidable amount of computation (especially for such an innocent looking problem), and my three coauthors deserve credit for doing the heavy-lifting. I’ve been told that there had been many late night marathon sessions in an MIT Building 2 classroom where they tore apart one inequality after another.

This is certainly not the end of the story, as many more interesting problems on the subject remain unsolved—my favorite one being the analogous problem for colorings, e.g., instead of counting the number of independent sets, how about we count the number of ways to color the vertices of the graph with 3 colors so that no two adjacent vertices receive the same color? (See my previous blog post for some more discussion.)

Anyway, here is the the answer to the question above. The number of independent sets in an {n}-vertex graph satisfying the above properties is at most {c^n}, where the optimal constant for {c} is {15^{4/63}\times  23^{1/21} \times 31^{1/28}=1.558\dots}, or, more helpfully, the maximum is attained by the following graph (in general, the theorem says that the maximizer is always a disjoint union of complete bipartite graphs):


Extremal regular graphs

This post is adapted from my new expository survey Extremal regular graphs: independent sets and graph homomorphisms.

The earliest result in extremal graph theory is usually credited to Mantel, who proved, in 1907, that a graph on {n} vertices with no triangles contains at most {n^2/4} edges, where the maximum is achieved for a complete bipartite graph with half of the vertices on one side and half on the other side. Much more is now known about the subject. While I initially encountered Mantel’s theorem as a high school student preparing for math contests, my first in-depth exposure to extremal graph theory was from taking a course by David Conlon during my year reading Part III at Cambridge (one can find excellent course notes on his website).

We seem to know less about sparse graphs (a general mantra in combinatorics, it seems). Let us focus on {d}-regular graphs, which are graphs where every vertex has degree {d}.


The cycle of length 4 has a total of 7 independent sets.

An independent set in a graph is a subset of vertices with no two adjacent. Many combinatorial problems can be reformulated in terms of independent sets by setting up a graph where edges represent forbidden relations.

Question. In the family of {d}-regular graph of the same size, which graph has the most number of independent sets?

This question was raised by Andrew Granville in the 1988 Number Theory Conference at Banff, in an effort to resolve a problem in combinatorial number theory, namely the Cameron–-Erdős conjecture on the number of sum-free sets. The question appeared first in print in a paper by Noga Alon, who proved an asymptotic upper bound and speculated that, at least when {n} is divisible by {2d}, the maximum should be attained by a disjoint union of complete bipartite graphs {K_{d,d}}.

Some ten years later, Jeff Kahn arrived at the same conjecture while studying a problem arising from statistical physics. Using a beautiful entropy argument, Kahn proved the conjecture under the additional assumption that the graph is already bipartite.

Fast forward another nearly ten years. In the summer of 2009, during my last summer as an undergraduate, I spent a fun and productive summer attending Joe Gallian’s REU in Duluth, MN (a fantastic program, by the way!), and there I showed that Kahn’s theorem can be extended to all regular graphs, not just bipartite ones.

Here is the theorem statement. We write {I(G)} to denote the set of independent sets in {G}, and {i(G) := |I(G)|} the number of independent sets in {G}.

Theorem (Kahn, Z.) If {G} is an {n}-vertex {d}-regular graph, then

\displaystyle i(G) \le i(K_{d,d})^{n/(2d)} = (2^{d+1} - 1)^{n/(2d)}.

In the survey, I provide an exposition of the proofs of these two theorems as well as a discussion of subsequent developments. Notably, Davies, Jenssen, Perkins, and Roberts recently gave a brand new proof of the above theorems by introducing a powerful new technique, called the occupancy method, inspired by ideas in statistical physics, and it already has a number of surprising new consequences.

On a personal level, I am pleased to see this topic gaining renewed interest. (Also, somewhat to my chagrin, my undergraduate paper, according to Google Scholar, still remains my most cited paper to date.)

I shall close this blog post with one of my favorite open problems in this area.

Let {c_q(G)} denote the number of proper {q}-colorings of {G}, i.e., coloring the vertices of {G} with {q} colors, so that no two adjacent vertices receive the same color.

Conjecture. If {G} is an {n}-vertex {d}-regular graph, and {q \ge 3}, then

\displaystyle c_q(G) \le c_q(K_{d,d})^{n/(2d)}.

I was pleased to learn from Will Perkins, who gave a talk at Oxford last week, that he, along with Davies, Jenssen, and Roberts, recently proved the conjecture for 3-regular graphs. The proof uses the occupancy method that they developed earlier. The method is reminiscent of the flag algebra machinery developed by Razborov some ten years ago for solving extremal problems in dense graphs. The new method can be seen as some kind of “flag algebra for sparse graphs”. I find this development quite exciting, and I expect that more news will come.


Graph homomorphisms from G to K_q correspond to proper colorings of vertices of G with q colors.

Upper tails for triangles in sparse random graphs

Eyal Lubetzky and I just finished and uploaded to the arXiv our new paper On the variational problem for upper tails of triangle counts in sparse random graphs. This paper concerns the following question:


The upper tail problem for triangles. What is the probability that the number of triangles in an Erdős-Rényi graph graph {\mathcal{G}_{n,p}} is at least twice its mean, or more generally, larger than the mean by a factor of {1 + \delta}?

This problem has a fairly rich history, and it is considered a difficult problem in probabilistic combinatorics. In a 2002 paper titled The infamous upper tail, Janson and Ruciński surveyed the progress on the problem and described its challenges (they also considered the upper tail problem more generally for any subgraph {H}, but we focus on triangles here). Research towards the solution of this problem has led to new techniques for bounding probabilities of rare events. Note that in contrast, the lower tail problem — finding the probability of getting too few triangles — is comparatively easier.

Here we consider the case when {p} tends to zero as {n} grows (the dense case, where {p} is fixed, leads to an interesting variational problem involving graph limits, as shown by a seminal work of Chatterjee and Varadhan — see my previous blog post). It is not too hard to describe what the answer should be in the sparse setting. Suppose we arbitrarily select a set of {k = \delta^{1/3}np} vertices and force it to be a clique, that is, we force all edges between these {k} vertices to be present in the random graph — the probability “cost” of this constraint is {p^{\binom{k}{2}} = p^{(\delta^{2/3}/2 + o(1))n^2p^2}} — the cliques gives us roughly an additional {\binom{k}{3} = (\delta + o(1))\binom{n}{3} p^3} triangles, thereby boosting the total number of triangles in the graph by a factor of {1 + \delta} from the mean, which is {\binom{n}{3}p^3}.

Let {t(G)} denote the triangle density in a graph {G}, i.e., the number of triangles in {G} as a fraction of the maximum possible number {\binom{n}{3}} of triangles. The above argument shows that the upper tail probability satisfies

\displaystyle \mathbb{P}(t(\mathcal{G}_{n,p}) \geq (1 + \delta)p^3) \geq \exp\Big(-(\tfrac12 \delta^{2/3} +o(1))n^2 p^2 \log(1/p)\Big).

Is this lower bound tight? That is, can we find a matching upper bound for the upper tail probability? This is precisely the infamous upper tail problem mentioned earlier. For a long time there was no satisfactory upper bound anywhere close to the lower bound. A breakthrough by Kim and Vu in 2004 gave an upper bound of the form {\exp(-c n^2p^2)}. However, this still leaves a missing factor of {\log(1/p)} in the exponent. This problem was finally settled in 2012 by Chatterjee and independently by DeMarco and Kahn. They showed that the exponent in the probability has order {n^2p^2 \log(1/p)}, thereby filling in the “missing log.”

Now we know the correct order in the exponent, what remains unknown is the constant in front of the {n^2p^2 \log(1/p)}. In a very recent breakthrough by Chatterjee and Dembo (the preprint was released just January this year), they provided a framework that reduces this large deviation problem to a certain variational problem, at least in the range when {p \geq n^{-\alpha}} for some explicit (though small) {\alpha > 0}. The variational problem is the natural one associated to this problem, and it is expected to hold for much sparser random graphs as well.

Eyal and I have been working on this variational problem for quite some time, starting with our previous paper where we studied the problem in the dense setting (constant {p}) and determined the phase diagram for replica symmetry. In our new paper, we solve this variational problem in the sparse regime. Combining the solution of this variational problem with the new results of Chatterjee and Dembo, this determines the correct asymptotic in the exponent of the upper tail probabilities, at least when {n^{-\alpha} \ll p \ll 1}. For fixed {\delta > 0}, we can now prove that

\displaystyle \mathbb{P}(t(\mathcal{G}_{n,p}) \geq 1 + \delta) = \exp\Big(-(C_\delta+o(1))n^2 p^2 \log(1/p)\Big).

where we determine the constant in the exponent to be

\displaystyle C_\delta = \min\Big\{\tfrac12 \delta^{2/3}, \tfrac13 \delta\Big\}.

We described earlier the story behind the {\tfrac12 \delta^{2/3}} term (by forcing a clique of size {\delta^{1/3}np}). Where does the other term, {\tfrac13 \delta}, come from? It turns out there is another way to get a lots of triangles relatively cheaply, at least when {p \gg n^{-1/2}}: forcing a set of {\ell = \frac13 \delta np^2} vertices to be connected to all other vertices. This has probability {p^{\ell(n - \ell)} = p^{(\delta/3 + o(1))n^2p^2}}. The first construction (forcing a clique) competes with the second construction (forcing a complete bipartite graph): the former is preferable when {\delta > 27/8} and the latter is preferable when {\delta < 27/8}. It turns out, as we show in our paper, that these are essentially the only constructions.

Recent developments on the Green-Tao theorem

There has quite a bit of activities around the Green-Tao theorem recently. In this blog post, I want to highlight some of these developments, focusing on the following papers, which were all posted on the arXiv in the past couple of months.

Relative Szemerédi theorem

Multidimensional Szemerédi theorem in the primes


We start the story with Szemerédi’s famous result.

Szemerédi’s theorem. Every subset of integers with positive density contains arbitrarily long arithmetic progressions.

Green and Tao proved their famous theorem extending Szemerédi’s theorem to the primes.

Green-Tao theorem. The primes contain arbitrarily long arithmetic progressions. In fact, any subset of the primes with positive relative density contains arbitrarily long arithmetic progressions.

An important idea in Green and Tao’s work is the transference principle. They transfer Szemerédi’s theorem as a black box to the sparse setting, so that it can be applied to subsets of a sparse pseudorandom set of integers (in this case, some carefully designed enveloping set for the primes).

Jacob Fox, David Conlon, and myself

Jacob Fox, David Conlon, and me at the Erdős Centennial conference in Budapest, Hungary, July 2013.

In my recent paper with David Conlon and Jacob Fox, we gave a new simplified approach to proving the Green-Tao theorem. In particular, we established a new relative Szemerédi theorem, which required simpler pseudorandomness hypotheses compared to Green and Tao’s original proof. Roughly speaking, a relative Szemerédi theorem is a result of the following form.

Relative Szemerédi theorem (roughly speaking). Let S be a pseudorandom subset of integers. Then any subset of S with positive relative density contains long arithmetic progressions.

The original proof in our paper followed the hypergraph approach, and in particular used the hypergraph removal lemma, a deep combinatorial result, as a black box. Subsequently, I wrote up a short six-page note showing how to prove the same result by directly transferring Szemerédi’s theorem, without going through hypergraph removal lemma. The former approach is more powerful and more general, while the latter approach is more direct and gives better quantitative bounds.

Next we shift our attention to higher dimensions. Furstenberg and Katznelson proved, using ergodic-theoretic techniques, a multidimensional generalization of Szemerédi’s theorem.

Multidimensional Szemerédi theorem (Furstenberg and Katznelson). Any subset of Zd with positive density contains arbitrary constellations.

Here a constellation means some finite set R of lattice points, and the theorem says that the subset of Zd contains some translation of a dilation of R.

(This result always reminds me of a lovely scene in the movie A Beautiful Mind where the John Nash character traces out shapes among the stars.)

"Pick a shape" - A Beautiful Mind

Pick a shape” – A Beautiful Mind

Tao proved a beautiful extension for the Gaussian primes.

Theorem (Tao). The Gaussian primes contain arbitrary constellations.

In that paper, Tao also made the following interesting conjecture: let P denote the primes. Then any subset of P × P of positive relative density contains arbitrary constellations. This statement can be viewed as a hybrid of the Green-Tao theorem and the multidimensional Szemerédi theorem. Recently this conjecture was resolved by Tao and Ziegler, and independently by Cook, Magyar, and Titichetrakun.

Multidimensional Szemerédi theorem in the primes. Let P denote the primes. Then any subset of Pd with positive density contains arbitrary constellations.

Terry Tao has written a blog post about this new result where he describes the ideas in the proofs (so I won’t repeat too much). Tao and Ziegler proved their result via ergodic theory. Their proof is about 19 pages long, and they assume as black boxes two powerful results: (1) Furstenberg and Katznelson’s multidimensional Szemerédi theorem (more precisely their equivalent ergodic-theoretic formulation) and (2) the landmark results of Green, Tao, and Ziegler on linear equations in primes (and related work on the Mobius-nilsequences conjecture and the inverse conjecture on the Gowers norm). Both these results are deep and powerful hammers.

In contrast, Cook, Magyar, and Titichetrakun proceeded differently as they develop a sparse extension of hypergraph regularity method from scratch, without assuming previous deep results. Though their paper is much longer, at 44 pages.

In a recent paper by Jacob Fox and myself, we give a new and very short proof of the same result, assuming the same tools as the Tao-Ziegler proof. Our paper is only 4 pages long, and it uses a very simple sampling argument described in two paragraphs on the first page of our paper (all the ideas are on the first page; the rest of the paper just contains the technical details spelled out). I invite you to read the first page of our paper to learn the very short proof of the theorem.

The Green-Tao theorem and a relative Szemerédi theorem

David Conlon, Jacob Fox, and I just uploaded to the arXiv our second joint paper, titled “A relative Szemerédi theorem.”

Before describing our work, let me to take a detour to reflect on some recent news in number theory.

The past couple weeks were filled with exciting developments on prime numbers. Yitang Zhang, in a breakthrough that caught the mathematical community by complete surprise, proved that there exists infinitely many pairs of primes with bounded gaps (specifically, less than 70 million). This is a giant leap of progress towards the famous twin primes conjecture, which claims that there exist infinitely pairs of primes differing by two.

By an amazing coincidence, in the same day that Zhang’s news became public, Harald Helfgott posted on the arXiv his paper which claims to prove the odd Goldbach conjecture, that every odd integer greater than 5 can be written as a sum of three primes. Previously this was known to be true for sufficiently large odd numbers, specifically those with at least about a thousand digits. It is possible to check the conjecture by computer for “small” cases, but 101000 is far from small. Helfgott’s work brought the “sufficiently large” down to a reasonable threshold, so that all the remaining small cases can indeed be verified by computer search.

Both results are exciting developments. However, we still seem far away from resolving two of the oldest problems about prime numbers. Zhang’s bound of 70 million probably could be brought down by more careful analysis, although getting it all the way down to 2 will probably require some really new ideas. As for Goldbach’s conjecture, there’s significant obstacle in getting down to two primes (the best result in this direction is due to Jingrun Chen), as all existing methods have severe limitations. But then again, the same could have been said about bounded gaps between primes before Zhang’s surprising work. Perhaps one day, we’ll be shocked again with a breakthrough on Goldbach, maybe even using techniques right in front of our eyes.


Ben Green and Terry Tao

Ben Green and Terry Tao

Back in 2004, the mathematical community received a surprise on another long-standing open problem on prime numbers: Ben Green and Terry Tao proved that the primes contain arbitrarily long arithmetic progressions.

An arithmetic progression is a sequence of equally spaced numbers. For example, 3, 5, 7 is a 3-term arithmetic progression of prime numbers. To find one with four terms, you need to look a bit further: 5, 11, 17, 23. At the time of Green and Tao’s proof, the longest such progression known was a sequence of 22 numbers, starting with 11410337850553 and with step size 4609098694200. This record has since been improved to 26. Here’s the description from Wikipedia:

As of April 2010, the longest known AP-k is an AP-26, found on April 12, 2010 by Benoãt Perichon on a PlayStation 3 with software by Jaroslaw Wroblewski and Geoff Reynolds, ported to the PlayStation 3 by Bryan Little, in a distributed PrimeGrid project.

There is a webpage with current records on primes in arithmetic progressions.

It was a long standing open conjecture that the primes should contain arithmetic progressions of every length. This was a folklore conjecture, with the first recorded instance in history dating back to 1770 by Lagrange and Waring. Green and Tao proved this conjecture, and their breakthrough is considered one of the greatest mathematical achievements of the twenty-first century. This work was an important part of Tao’s 2006 Fields Medal award.


I was still in high school when the Green-Tao theorem was announced. The news spread quickly online, and I soon heard of it through my math competition online community. I remember being quite excited about the news. It was a quintessential mathematical breakthrough. The statement of the problem could be easily understood, yet the solution involved very deep mathematics. I fell in love with the statement of the Green-Tao theorem. It became one of my favorite mathematical results, and I aspired to one day understand the mathematics behind it.

Now, nine years later, as a graduate student, I find it deeply personally fulfilling to be working on a problem central to the Green-Tao theorem. Our latest paper (joint with Conlon and Fox) strengthens the main technical result in the work of Green and Tao.

Endre Szemerédi

Endre Szemerédi

To describe this result, I need to first tell you about another important mathematical breakthrough by Endre Szemerédi dating back to the early 1970’s. Last year, Szemerédi received the Abel Prize, one of the highest lifetime achievement awards in mathematics. Szemerédi is perhaps most famous for his result, commonly referred to as Szemerédi’s theorem, which says that every subset of integers with positive density contains arbitrarily long arithmetic progressions. So, if S is a sufficiently large subset of the integers, “large” meaning that it contains, say, at least 0.1% of the first N positive integers for sufficiently large N, then S necessarily contains an arithmetic progression of every length.

Szemerédi’s theorem is a very deep result, albeit again with a seemingly very innocent statement. It was conjectured by Erdős and Turán in the 1930’s, and remained open for decades. Since Szemerédi’s breakthrough, several other proofs have been discovered. None of the proofs are easy but all of them have deep insights and have spawned into very rich and active areas of mathematical research.

I was personally exposed to Szemerédi’s theorem while at Cambridge University. I spent a year there right after finishing undergrad and took part in what is colloquially known as Part III of the Maths Tripos. One of the components of the program was writing an extended expository essay, and I wrote mine about Szemerédi’s theorem under the supervision of Ben Green. Ben gave me a very high score for the essay. I’ve put the essay on my website, and several people have told me that they found it to be helpful (including one who was preparing a popular article on Szemerédi after his Abel Prize award). Though, to be honest, looking back, perhaps now I would be a bit embarrassed to read that essay again myself, since I wonder how much I actually understood while I was writing it up at the time.

While Szemerédi’s theorem is a powerful result, it is not enough to draw any conclusions about prime numbers. Szemerédi’s theorem only works for sets of integers with positive density. The primes, on the other hand, have density diminishing to zero. Indeed, the Prime Number Theorem tells us that between 1 and N, approximately 1 in ln N fraction of the numbers are prime. This ratio, 1/ln N, diminishes to zero as N grows to infinity. So Szemerédi’s theorem doesn’t work here.

The primary innovation of Green and Tao is that they came up with a relative version of Szemerédi’s theorem. To overcome the problem that the primes have zero density, they find a slightly larger set of integers where the primes can sit inside as a subset of positive relative density. This larger host set, roughly speaking, is the set of “almost primes,” which consists of numbers with few prime divisors. They found a way to transfer Szemerédi’s theorem to this relative setting, showing that if we start with a set with some random-like characteristics (in this case, the almost primes), then any subset of it with positive density relative to the host set must necessarily contain long arithmetic progressions.

The Green-Tao paper splits into two parts. In the first part, which is their main technical contribution, they establish a relative Szemerédi theorem for subsets of pseudorandom sets of integers. Their method was heavily influenced by the work of Fields Medalist Timothy Gowers, who revolutionized the field of additive combinatorics by inventing a far-fetching extension of Fourier analysis (also known in this context as the Hardy-Littlewood circle method) to give a novel proof of Szemerédi’s theorem. The second part of Green and Tao’s paper shows that the almost primes act as a suitable host set by verifying the required pseudorandomness conditions. Most of the number theoretic input to their work were credited to the works of Goldston and Yıldırım, which later led to a spectacular breakthrough in the problem of small gaps in primes by Goldston, Pintz, and Yıldırım (which relates back to Zhang’s twin primes breakthrough mentioned at the beginning of this blog post).

The pseudorandomness conditions required in the Green-Tao method are rather involved. It assumes that the host pseudorandom set satisfies a “linear forms condition” as well as a “correlations condition.” Both conditions are essential to Green and Tao’s method of establishing their relative Szemerédi’s theorem. They show that the almost primes satisfy these pseudorandomness conditions, thereby proving the old conjecture about progressions in primes. However, the complicated pseudorandomness hypotheses, while adequate for this application, seem rather contrived and somewhat unsatisfying. This leads to the next question, which has been repeatedly asked since the Green-Tao work: does a relative Szemerédi theorem hold under more natural pseudorandomness hypotheses?

In our new paper, we show that the answer is yes! We prove a relative Szemerédi theorem under a very simple and natural linear forms condition. What we need to assume is a small subset of what Green and Tao assume in their work. This new result provides not only a brand new (and simpler, as we believe) alternative approach to the proof of the Green-Tao theorem, but also, more broadly speaking, a new method for understanding sparse pseudorandom structures. The Green-Tao method has had a large number of applications since its inception almost a decade ago. With our new perspective, perhaps one can go even further.

Sphere packing bounds via spherical codes

Henry Cohn and I just uploaded to the arXiv our paper “Sphere packing bounds via spherical codes.” [Update: Henry gave a wonderful talk about our paper at the IAS. A video of the talk is available here.]

KeplerConjectureWhat’s the most space-efficient way to arrange a collection of identical balls? This is known as the sphere packing problem. It is a very difficult problem with a long and interesting history. This problem in 3-dimensions is known as Kepler conjecture, which says that the the face-centered cubic formation does best. This is basically how grocers stack oranges. The seemingly innocent Kepler conjecture turned out to be extremely difficult, and it was only solved in the late 1990’s by Thomas Hales who gave a very complex proof involving massive computer-aided calculations. I’ve included a few article links at the end of this blog post on the history and background of the sphere packing problem.

Now, what about sphere packing in higher dimensions? Unfortunately, we know very little about what happens beyond three dimensions. No proof of optimality is known in any higher dimension, and there are only a few dozen dimensions in which there are even plausible conjectures for the densest packing. In dimensions 8 and 24 there are upper bounds that are extremely close to the conjectured optima, thanks to the works of Cohn, Elkies, and Kumar [1,2,3] (dimensions 8 and 24 are somehow special because of the existence of highly symmetric lattices known as the {E_8} lattice in dimension 8 and the Leech lattice in dimension 24). However, in most dimensions we must be content with much cruder bounds.

The current state of art for sphere packing density upper bounds is more or less as follows:

  • In dimensions 1, 2, 3, the exact upper bound is known. The result for dimension 3 is due to Hales.
  • In low dimensions, namely 4 to 42, Cohn and Elkies improved previous record by Rogers, although there were some recent improvements in dimensions 4,5,6,7,9 by de Laat, Filho, and Vallentin using techniques of semidefinite programming.
  • In all high dimensions, namely 43 and above, the best bounds are due to Kabatiansky and Levenshtein in 1978 and have not been improved since then.

The purpose of our of paper is fourfold:

  1. We give a small improvement over the 1978 bounds of Kabatiansky and Levenshtein in high dimensions by giving a simple modification of their geometric argument relating spherical codes to sphere packings.
  2. Kabatiansky and Levenshtein derived their bounds by first formulating a linear program for proving upper bounds on spherical codes. Cohn and Elkies found a more direct approach to bounding sphere packing densities, with no need to consider spherical codes. However, despite the excellent performance in low dimensions, the asymptotic behavior of the Cohn-Elkies bound is far from obvious and it has been unclear whether it improves on, or even matches, the Kabatiansky-Levenshtein bound asymptotically. In our paper, we show that in every dimension {n}, the Cohn-Elkies linear program can always match the Kabatiansky-Levenshtein approach. This further demonstrates the power of the linear programming bound for sphere packing.
  3. We prove an analogue of the Kabatiansky-Levenshtein bound in hyperbolic space. The resulting bound is exponentially better than the best bound previously known in hyperbolic space.
  4. We develop the theory of hyperbolic linear programming bounds and prove that they too subsume the Kabatiansky-Levenshtein approach. Packing in hyperbolic space is much more difficult to handle than in Euclidean space, primarily because the volume of an expanding ball grows exponentially with its radius (instead of polynomially as in the case of Euclidean space), so we that cannot neglect boundary fluctuations. In fact, it is a non-trivial matter to even define the density of a packing (there are examples of packings for which any reasonable definition of density should yield two different answers).

Further reading

Some articles on the history and background of the sphere packing problem:

Popular audience:

General mathematical audience:

Energy-minimizing error-correcting codes

Henry Cohn and I just uploaded to the arXiv our paper “Energy-minimizing error-correcting codes.”

Hamming cubeGeometry has long played a key role in coding theory, starting with the work of Hamming: binary codes can be viewed as packings of Hamming balls in a discrete cube. This framework provides a powerful analogy between discrete and continuous packing problems, which has been extensively developed and remains an active research topic. In our paper, we extend the analogy to a much broader relationship between coding theory and discrete models of physics. Of course, physics is related to coding theory in many ways, ranging from connections between spin glasses and codes to the statistical physics of belief propagation and other applications of graphical models to coding theory. Applications of physics to coding theory typically focus on the limit as the block length tends to infinity. Instead, in this paper we show that certain classical codes are exact ground states of natural physics models.

In addition to extending the analogy with continuous packing problems, our results can be thought of as addressing a philosophical problem. Many classical codes—such as Hamming, Golay, or Reed-Solomon codes—remain very popular, despite the many other good codes that have been found. Why should this be? One obvious answer is that these codes are particularly beautiful and useful, especially given the simplicity of their constructions. Another is that they were discovered early in the development of coding theory and had a chance to cement their place in the canon. We propose a third explanation: a code is most useful if it is robust, in the sense that it optimizes not just one specific measure of quality, but rather a wide range of them simultaneously. We will prove that these classical codes have a rare form of robustness that we call universal optimality, based on an analogy with continuous optimization problems.

Thomson modelTo see this analogy, recall the following classic problem: given {n} particles on a sphere interacting via some mutually repelling force (the Thomson model for electrons is a good example of this), in what configurations would the particles arrange themselves? The configuration(s) of greatest interest is the ground state, which is the one possessing the least potential energy.

This then leads to the following fundamental problem in extremal geometry: given some metric space (e.g. sphere) and an arbitrary potential function based on the distance between pairs of points, for any positive integer {N}, how should {N} points arrange themselves to minimize the total potential energy of the system?

One might expect that difference choices of potential energies might lead to different ground state configurations—and this is usually the case. However, there are some highly symmetric configurations which are highly robust in the sense that they minimize not just a single potential function, but a broad class of potential functions.

What kind of potential functions should we be looking at? For example, the electric potential function in {{\mathbb R}^3} is {f(r) = 1/r}. The natural generalization to {{\mathbb R}^n} is {f(r) = 1/r^{n-2}}. However, we are also interested in more general potentials. Ideally, the potential function should possess certain properties. First, it should be decreasing with distance (since we’re interested in repelling forces; the ground state with an attractive force isn’t so interesting, as all particles would collapse into one point). The potential function should also be convex, since the effect of the force should diminish with distance. We can express these two conditions as {f' \leq 0} and {f'' \geq 0} respectively. Let us extrapolate from these two conditions and impose similar conditions on higher order derivatives, namely, {f''' \leq 0}, {f^{(4)} \geq 0}, etc. Note that all power laws {f(r) = r^{\alpha}} for {\alpha < 0} satisfy these conditions.

The work of Cohn and Kumar studied precise this class potential functions, which they called completely monotonic, for points on a sphere in {{\mathbb R}^n} (actually, for technical reasons, they consider functions {f} of the square of the distance between pairs of points). They studied configurations of points on a sphere which minimize all completely monotonic potential function. Examples of such configurations include the regular simplex, the cross polytope, the icosahedron, the 600-cell, roots of the {E_8}-lattice, and the minimal vectors of the Leech lattice. These beautiful and highly symmetric configurations all have very robust energy minimization properties.

In our current work, we analyze error-correcting codes using this perspective of energy minimization. Binary error-correcting codes can be thought of set of particles on a high-dimensional cube. We are interested in finding out which error-correcting codes minimize a broad class of potential energies? The class of potential function is the discrete analogue of the completely monotonic functions described above, with the conditions on derivatives replaced by conditions on successive finite differences.

As mention in the beginning, we show that many classical codes have robust minimization properties, which translate into good performance according to a broad range of measures. For example, such codes minimize the probability of an undetected error under the symmetric channel, and it also has interesting consequences for other decoding error probabilities.

Our main technical tool for bounding energy is the linear program developed by Delsarte, which was originally used to bound the size of codes given their minimum distance. We will call a code LP universally optimal if its universal optimality follows from these bounds. One of our key results is that LP universal optimality behaves well under duality, thereby allowing us to apply our criteria to many classes of codes.

One result we found particularly surprising is that LP universally optimal codes continue to minimize energy even after we remove a single codeword. We know of no analogue of this property in the continuous setting. This property also implies structural properties, namely that every LP universally optimal code is distance regular, i.e., for each distance, every codeword has the same number of codewords at that distance.

On replica symmetry of large deviations in random graphs

Eyal Lubetzky and I just uploaded to the arXiv our new paper “On replica symmetry of large deviations in random graphs.” In this paper we answer the following question of Chatterjee and Varadhan:

Question. Fix {0<p<r<1}. Let {n} be a large integer and let {G} be an instance of the Erdős-Rényi random graph {\mathcal G(n,p)} conditioned on the rare event that {G} has at least as many triangles as the typical {\mathcal G(n,r)}. Does {G} look like a typical {\mathcal G(n,r)}?

Here the Erdős-Rényi random graph {\mathcal G(n,p)} is formed by taking {n} vertices and adding every possible edge independently with probability {p}. Here “look like” means close in cut-distance (but we won’t give a precise definition in this blog post). In this case, saying that {G} is close to {\mathcal G(n,r)} is roughly equivalent to saying that every not-too-small subset of vertices (at least constant fraction in size) of {G} induce a subgraph with edge density close to {r}.

Another way to phrase the question is: what is the reason for {G} having too many triangles? Is it because it has an overwhelming number of edges uniformly distributed, or some fewer edges arranged in a special structure, e.g., a clique.

Via a beautiful new framework by Chatterjee and Varadhan for large deviation principles in {\mathcal G(n,p)}, we give a complete answer to the above question.

The answer, as it turns out, depends on {(p,r)}. See the plot below. For {(p,r)} in the blue region, the answer is yes, and for {(p,r)} in the red region, the answer is no.

Does {G} looks like a {\mathcal G(n,r)}?

The phase transition behavior has already been observed previously by Chatterjee and Varadhan, but the determination of the exact precise phase boundary is new. Borrowing language from statistical physics, the blue region where the conditional random graph is close to {\mathcal G(n,r)} is called the replica symmetric phase, and the red region is called the symmetry breaking phase. Note that in the left part of the plot, as we fix a small value of {p}, the model experiences a double phase transition as {r} increases from {p} to {1} — starting first in the blue phase, then switches to the red phase, and then switches back to the blue phase.

More generally, our result works for any {d}-regular graph in replace of triangles. The boundary curve depends only on {d}, and they are plotted below for the first few values of {d}. In particular, this means that large deviation for triangles and 4-cycles share the same phase boundary. A pretty surprising fact! We also consider the model where we condition on the largest eigenvalue of the graph being too large, and the phase boundary also turns out to be the same as that of triangles.

We also derive similar results for large deviations in the number of linear hypergraphs in a random hypergraph.

The phase boundary for {d}-regular graphs

Exponential random graphs

We also studied the exponential random graph model. This is a widely studied graph model, motivated in part by applications in social networks. The idea is to bias the distribution of the random graph to favor those with, say, more triangles. This model has a similar flavor to the model considered above where we condition on the random graph having lots of triangles.

We consider a random graph {G} on {n} vertices drawn from the distribution

\displaystyle p_\beta(G) \propto \exp\left(\tbinom{n}{2}(\beta_1 t(K_2, G) + \beta_2 t(K_3, G))\right)\,,

where {t(K_2, G)} and {t(K_3, G)} are the edge density and the triangle density of {G}, respectively. When {\beta_2 = 0}, this model coincides with the Erdős-Rényi model {\mathcal G(n,p)} with some {p} depending on {\beta_1}. We only consider the case {\beta_2 > 0}, which represents a positive bias in the triangle count.

As shown by Bhamidi, Bresler and Sly and Chatterjee and Diaconis, when {n} is large, a typical random graph drawn from the distribution has a trivial structure — essentially the same one as an Erdős-Rényi random graph with a suitable edge density. This somewhat disappointing conclusion accounts for some of the practical difficulties with statistical parameter estimation for such models. In our work, we propose a natural generalization that will enable the exponential model to exhibit a nontrivial structure instead of the previously observed Erdős-Rényi behavior.

Here is our generalization. Consider the exponential random graph model which includes an additional exponent {\alpha>0} in the triangle density term:

\displaystyle p_{\alpha,\beta}(G) \propto \exp\left(\tbinom{n}{2}(\beta_1 t(K_2, G) + \beta_2 t(K_3, G)^\alpha)\right) \, .

When {\alpha \geq 2/3}, the generalized model features the Erdős-Rényi behavior, similar to the previously observed case of {\alpha = 1}. However, for {0< \alpha < 2/3}, there exist regions of values of {(\beta_1, \beta_2)} for which a typical random graph drawn from this distribution has symmetry breaking, which was rather unexpected given earlier results. For example, we know that there is symmetry breaking in the shaded regions in the plots below. (The blue curve indicates a discontinuity in the model which we won’t discuss in this blog post.)

Symmetry breaking in the new exponential graph model

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The critical window for the classical Ramsey-Turán problem

[Update: Po-Shen Loh gave a wonderful talk in Banff about our paper. Here is the video.]

Jacob Fox, Po-Shen Loh and I recently uploaded to the arXiv our new paper “The critical window for the classical Ramsey-Turán problem.” This paper revisits the following celebrated Ramsey-Turán result first proved by Szemerédi in 1972.

Theorem 1 (Szemerédi 1972) For every {\epsilon > 0}, there exists a {\delta > 0} such that any graph on {n} vertices containing no 4-clique and no independent set of size greater than {\delta n} has at most {(1/8 + \epsilon)n^2} edges.

Turán’s theorem tells us that if we just want to construct a {K_4}-free graph on {n} vertices with the largest possible number of edges, then we should divide the {n} vertices into three nearly-equal parts and put in all possible edges across parts. This yields roughly {n^2/3} edges. However, this graph contains very large independent sets with {n/3} vertices. So the independent set restriction (which gives the “Ramsey” part of Ramsey-Turán) rules out this construction and forces the optimal graphs to take on a very different structure.

At the time of Szemerédi’s 1972 result, most people believed that a {K_4}-free graph with largest independent set of size {o(n)} has to be much more sparse, perhaps with only {o(n^2)} edges. So it was a surprise when, four years later, Bollobás and Erdős gave a clever geometric construction, based on the isoperimetric inequality for the high dimensional sphere, that showed that the bound given by Szemerédi is essentially optimal.

Theorem 2 (Bollobás and Erdős 1976) There exists a {K_4}-free graph on {n} vertices with largest independent set of size {o(n)} and having {(1/8 - o(1))n^2} edges.

Bollobás and Erdős asked what happens in the critical window, when the number of edges is about {n^2/8}. This problem has received considerable attention (e.g., it was featured in a paper by Erdős from 1990 titled “Some of my favourite unsolved problems”). One of the difficulties here is that Szemerédi’s original proof used the regularity lemma, which is a powerful tool but one that unfortunately gives very poor parameter dependencies. To obtain results for the Ramsey-Turán problem at the desired precision, a regularity-free proof is needed. We give a new proof of the classical Ramsey-Turán result that avoids the use of regularity. This allows us to obtain much better and in fact nearly-optimal dependencies for the classical Ramsey-Turán problem in the critical window near {n^2/8} edges, and solve several longstanding open problems in this area.

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