I’m happy to announce our new paper Equiangular lines with a fixed angle joint with four MIT coauthors: Zilin Jiang, Jonathan Tidor, Yuan Yao, and Shengtong Zhang.

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Zilin is an Instructor (postdoc), Jonathan is my PhD student who just finished his second year, and Yuan and Shengtong are undergraduates who just finished their second and first respective years.

Our paper solves a longstanding problem in discrete geometry concerning equiangular lines, which are configurations of lines in space that are pairwise separated by the same angle. How many such lines can exist simultaneously in a given dimension?

We determine, for every fixed angle, the maximum number of equiangular lines with the given angle in high dimensions.

Our paper builds on a long sequence of earlier ideas. There were some important progress on this problem by Igor Balla, Felix Dräxler, Peter Keevash, and Benny Sudakov, as featured in this earlier Quanta Magazine article written for a lay audience.

In the end we finished off the problem in a clean and crisp manner, in a 10-page paper with a self-contained proof.

It has been known that the study of equiangular lines is closed linked to spectral graph theory, which seeks to understand graphs via their spectrum (i.e., eigenvalues). We prove our main result via a new insight in spectral graph theory, namely that a bounded degree graph must have sublinear second eigenvalue multiplicity.

I just finished a new paper, Impartial digraphs, coauthored with Yunkun Zhou, who just completed his undergraduate studies at MIT and will be moving to Stanford to start his PhD this fall.

The problem (along with the conjecture that we proved) was proposed by Jacob Fox, Hao Huang, and Choongbum Lee. Ron Graham had also told me about this problem with much enthusiasm during tea time one afternoon at the Berkeley Simons Institute a couple of years ago, after learning about it from Jacob.

I really enjoyed working on this paper with Yunkun. The problem is natural and simple to state, the answer appears somewhat unexpected (at least initially), and the solution ended up being quite tricky yet clean, employing a variety of tools in combinatorics, ranging from analytic methods (graph limits), algebraic considerations (polynomial identities and factorization), as well basic combinatorial techniques such as bijections.

We are going to talk about directed graphs (which graph theorists like to abbreviate as digraphs). These are graphs where every edge is given an orientation.

A tournament is a directed graph where there is a directed edge between every pair of vertices (think of it as recording the outcome of a round-robin tournament where every player plays against every other player).

Given a directed graph H (think of it as a pattern), we can ask: how many different copies of the pattern H appear in a given tournament?

Look at the following 4-vertex tournament H. It has the following curious property: all tournaments of a fixed size contains the same number of copies of H, no matter how the edges in the tournament are oriented.

Hmm, why is this true? And are there other directed graphs H with the same property, namely that all tournaments of a given size contain the same number of copies of H?

In the good old mathematical tradition, we came up with a name for directed graphs H with this curious property: we called them impartial (as in: impartial judges of a tournament).

Okay. So why is the above directed graph impartial? Are there any other impartial directed graphs? What are all the impartial directed graphs?

Here is a simple example of an impartial digraph:

Here is another one:

Indeed, given any 10-vertex tournament, we can count the number of copies of the first edge (it doesn’t depend on the tournament), and then count the number of copies of the second edge among the remaining 8 vertices (it still doesn’t depend on the tournament).

Once these two edges are placed in a tournament, there are two ways to join their tips, but they result in identical patterns (one flipped from the other):

It must then follow that this directed graph is impartial as well:

We can follow the same logic and continue this procedure, iterate, and build up even larger impartial directed graphs. Each step, take two copies of the previous tree (keeping the same edge orientations), and then add a new edge between a twin pair of vertices.

We call any directed graph that can building up as above a recursively bridge-mirrored tree.

The same argument as earlier shows that every recursively bridge-mirrored tree is impartial.

Are there other impartial digraphs? Did we miss any?

It turns out, there are actually no others. We have found them all, and that is the main result of our paper:

Theorem. A directed graph is impartial if and only if every component is recursively bridge-mirrored trees.

Just a word about how about this problem is related to a bigger picture. It is related to a number of other graph homomorphism inequalities that are central in extremal graph theory. It can be considered as the “equality case” for a directed analog of an open problem known as Sidorenko’s conjecture. See our paper for some further discussions and open problems.

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.

Here are some highlights of our new results.

1. Graph colorings. Among d-regular graphs, which graph has the most number of proper q-colorings, exponentially normalized by the number of vertices of G?

We show that the extremal graph is the complete bipartite graph K_d,d (or disjoint unions thereof, as taking disjoint copies does not change the quantity due to the normalization).

2. Graph homomorphisms. Among d-regular graphs, which triangle-free graph G has the most number of homomorphisms into a fixed H, again exponentially normalized by the number of vertices of G?

We show that the extremal graph G is the complete bipartite graph K_d,d.

The triangle-free hypothesis on G is best possible in general. For certain specific H, such as K_q, corresponding to proper q-colorings as earlier, the triangle-free hypothesis can be dropped. It remains a wide open problem to determine for which H can one drop the triangle-free hypothesis on G.

3. Partition functions of Ferromagnetic spin models. Among d-regular graphs, which graph maximizes the log-partition function per vertex of a given Ferromagnetic spin model (e.g., Ising model)?

We show that the extremal graph is the clique K_d+1.

For each setting, we establish our results more generally for irregular graphs as well, similar to our earlier work on independent sets.

Our results can be interpreted as a reverse form of the inequality in Sidorenko’s conjecture, an important open problem in graph theory stating a certain positive correlation on two-variable functions.

One can also view our results as a graphical analog of the Brascamp-Lieb inequalities, a central result in analysis.

This paper resolves one of my favorite open problems on this topic (the number of graph colorings). It also points us to many other open problems. Let me conclude by highlighting one of them (mentioned in #2 earlier). I’ll state it in a simpler form for d-regular graphs, but it can be stated more generally as well.

Open problem. Classify all H with the following property: among d-regular graphs G, the number of homomorphisms from G to H, exponentially normalized by the number of vertices of G, is maximized by G = K_d,d.

Our results show that H = K_q works, even when some of its vertices are looped. Generalizing this case, we conjecture that all antiferromagnetic models H have the same property (though we know that this would not be the complete class, even if true).

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 such that every -vertex graph satisfying the above properties has at most independent sets?

In other words, letting denote the number of independent sets and the number of vertices of , maximize the quantity over all graphs 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 -vertex graph satisfying the above properties is at most , where the optimal constant for is , 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):

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 is at least twice its mean, or more generally, larger than the mean by a factor of ?

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 , 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 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 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 — the cliques gives us roughly an additional triangles, thereby boosting the total number of triangles in the graph by a factor of from the mean, which is .

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 of triangles. The above argument shows that the upper tail probability satisfies

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 . 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 , 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 . 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 for some explicit (though small) . 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 ) 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 . For fixed , we can now prove that

where we determine the constant in the exponent to be

We described earlier the story behind the term (by forcing a clique of size ). Where does the other term, , come from? It turns out there is another way to get a lots of triangles relatively cheaply, at least when : forcing a set of vertices to be connected to all other vertices. This has probability . The first construction (forcing a clique) competes with the second construction (forcing a complete bipartite graph): the former is preferable when and the latter is preferable when . It turns out, as we show in our paper, that these are essentially the only constructions.

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

• D. Conlon, J. Fox and Y. Zhao, A relative Szemerédi theorem.
• Y. Zhao, An arithmetic transference proof of a relative Szemerédi theorem.

Multidimensional Szemerédi theorem in the primes

• T. Tao and T. Ziegler, A multi-dimensional Szemerédi theorem for the primes via a correspondence principle.
• B. Cook, Á. Magyar and T. Titichetrakun, A multidimensional Szemerédi theorem in the primes.
• J. Fox and Y. Zhao, A short proof of the multidimensional Szemerédi theorem in the primes.

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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).

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 Z^d 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 Z^d 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.)

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 P^d 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.