At this point, we are finalizing a draft of a paper by D.H.J. Polymath. Like Aubrey’s original paper, this will be submitted for publication in Geombinatorics.

This post concludes the Polymath16 project. Of course, we anticipate that folks will continue to make progress on this problem on their own. If you’d like to update the Polymath16 community about your progress, feel free to comment on this post.

I asked Aubrey to offer a few words to reflect on our project:

The Polymath 16 project recently celebrated 1000 days of age. It had two main initial goals: to find 5-chromatic unit-distance graphs in the plane with fewer vertices than the 1581-vertex example that I had published in early 2018, and to identify proofs of that required less computer assistance than my construction needed (ideally none at all). Many ancillary goals were also mentioned at the outset.

By all standards, the project has been an immense success. The record for the smallest graph was progressively improved, including several times by Marijn Heule, and the present record is 509, achieved by Jaan Parts. Jaan also has the distinction of cracking the other main challenge: a few months ago he unveiled a proof of that he described only as “human-verifiable,” but that is too modest, because his method is certainly usable to create a proof from scratch without any computational assistance in no more than a week or two (for someone sufficiently diligent and non-error-prone!). Jaan has, in fact, been the most prolific of the contributors to the project over the years, having made numerous other contributions, some on his own and some in Polymath-esque rapid-fire collaboration with others.

Perhaps the most satisfying aspect of the project, to me, is its quite remarkable success in attracting amateur mathematicians like myself (of whom Jaan is one). At least half a dozen of us have been significant contributors over these three years, working side by side with numerous professionals including Terry Tao himself.

Given the diversity of Hadwiger-Nelson-type problems that we have investigated, it was always implausible that all our results would be gathered into a single publication. This was especially the case in view of the proportion (more than half) of our advances that were achieved by a single individual with little or no collaboration at the blog, and which were thus more properly published under that person’s name alone. However, one clear exception emerged: the set of questions concerning bounds on the sizes of circular disks and infinite strips of different chromatic number. It has turned out that at least five of us made major contributions to one or more such question, so this was the natural focus of what has just become the first (and, we must now anticipate, only) paper arising from Polymath 16 whose sole listed author is the canonical, fictional D.H.J. Polymath. Like several of the other reports, and like my original proof of , it will appear in the journal Geombinatorics, whose editor-in-chief Alexander Soifer was, via his timeless 2009 book, responsible for inspiring me to work on the Hadwiger-Nelson problem in the first place.

The energy that Polymath 16 has evoked will not dissipate easily. We plan to leave this blog open for further discussions and reports – which will surely emerge as time passes, since many fascinating Hadwiger-Nelson-type questions remain wide open in spite of our efforts. Let us hope that the merry band who have progressed this far will continue to grow and to make yet further inroads into this truly captivating family of questions.

]]>We’re kicking off the spring semester on January 26 with a talk from Steve Flammia on recent progress on Zauner’s conjecture.

Click here for the Zoom link and to sign up for the mailing list.

]]>Consider a weighted and edge-colored bipartite graph consisting of

- disjoint vertex sets and ,
- an edge set ,
- an edge coloring , and
- a complex vertex weighting .

This graph provides a combinatorial abstraction of a quantum mechanical system. Specifically, denotes a set of photon sources, is the set of optical output paths, is the set of all possible photons, gives the mode number of photon , and the complex weights allow one to model quantum interference between the photon sources. Consider the following experiment: Place a detector at each , and let each randomly produce either correlated photons or no photons. We are interested in the event that all detectors simultaneously detect exactly one photon. The mode numbers of these photons can be viewed as a vertex coloring , and this coloring reveals some information about which sources produced which photons. Specifically, we say is **-consistent** if

- the neighborhoods form a partition of , and
- for every and every , it holds that .

In quantum optics, we cannot directly access the mode numbers of the photons, and in fact, we obtain a superposition of various possibilities. This superposition forms something called a *Greenberger–Horne–Zeilinger (GHZ) state* if satisfies an additional *monochromatic* property, defined below. Let denote the set of all -consistent , and define the **weight** of a coloring of to be

By convention, implies . We say is **monochromatic** if the weight of every coloring of takes the form

Here, denotes the image of the map . In words, is monochromatic if indicates whether gives every member of the same color in the palette . If is monochromatic, then the corresponding quantum mechanical system produces GHZ states, and the dimensionality of the state space is the size of the color palette .

In practice, one is interested in producing high-dimensional states with a given number particles, and the most practical of photon sources creates only two photons. For these reasons, we are particularly interested in the supremum of over all monochromatic such that and for every . Let denote this supremum. Notice that the constraint on implies that is monochromatic only if is even. For such , Mario poses the following conjecture:

Excitingly, there are two prizes associated with this conjecture. First, there’s a 3000-Euro prize for its resolution (i.e., for either a proof or a disproof). In addition, there’s a 1000-Euro best-paper award offered for the best results obtained in this vein (to be judged by Mario). See this page for more details.

To explain the source of Krenn’s Conjecture, we point out that “half” of the conjecture is known to be true; that is, the inequality holds, while the inequality is open for every with . We obtain this lower bound on by restricting our attention to for which

- for every and , and
- for every .

This restriction is convenient, since we can capture using the edge-colored multigraph with vertex set , edge set , assignment defined by , and edge coloring defined by for every . For such , the corresponding is monochromatic if and only if the following hold simultaneously:

- for every color , there exists a unique perfect matching of such that , and
- for every perfect matching of , it holds that is a singleton set.

This reduces our problem to finding multigraphs whose perfect matchings are all disjoint, since one may color edges in the th perfect matching with , and then color any remaining edges with . Let denote the supremum of for which there exists a multigraph on vertices with a total of perfect matchings, all of which are disjoint. Then the above discussion gives

Notice we can already conclude for every by taking to be a -factor. In the special case where , one may draw arbitrarily many edges between the two vertices to obtain , i.e., . It remains to consider with . In this case, the multigraph we seek can be taken to be a simple graph without loss of generality (i.e., multiple edges are not helpful here), and so in this language, Ilya Bogdanov proved that equals the right-hand side of Krenn’s Conjecture. The case corresponds to the -factorization of , while the remaining cases correspond to the -factorization of :

In order to determine whether this lower bound is sharp, one must consider more general choices of . In particular, do non-unit weights on allow for a larger color palette? Interestingly, Mario has an example of with , , , and for every . However, this graph is only monochromatic in the limit as , so this does not quite prove that is at least .

]]>Click here for a draft of my lecture notes.

I will update the above link periodically. Feel free to comment below.

**Update #1: **Added two sections to Chapter 1.

**Update #2: **Updated Chapter 1 and started Chapter 2.

**Update #3: **Added a section to Chapter 2.

**Update #4: **Added another section to Chapter 2.

**Update #5: **Added another section to Chapter 2.

**Update #6: **Added Chapter 3 and started Chapter 4.

**Update #7: **Updated Section 4.1.

**Update #8: **Added Section 4.2.

First, let denote the set of matrices in such that

Here, denotes conjugate transpose, denotes entrywise squared modulus, denotes the identity matrix, and denotes the all-ones matrix. In words, the columns of form an **equiangular tight frame (ETF)** for of size .

One particularly interesting family of ETFs are of the form . In the quantum information theory community, these are known as **symmetric, informationally complete positive operator–valued measures**; the modern abbreviation for this mouthful is **SIC**. In his PhD thesis, Zauner predicted that SICs exist for every , but to date, they are only known to exist for finitely many . Almost all existing SICs are constructed by spinning a seed vector with a representation of the Weyl–Heisenberg group, and the entries of the seed vector are expressible in radicals. Frustratingly, the complexity of this description appears to grow like , which suggests that this is not the best representation for these seed vectors.

In 2018, Gene discovered an alternative that seems to be the “correct” representation. Specifically, the squares of the entries of appear to be obtained by applying a Galois automorphism to the Stark units of an abelian extension of with ramification at the primes dividing and at one infinite place. In the case when is prime, Gene gives a precise recipe to construct the entries of from their squares.

These equiangular tight frames exhibit field structure that is completely different from other known constructions. In general, given , the triple products with form an invariant over the left-action of and the right-action of . For all known ETF constructions other than SICs, either the ETF belongs to a continuum of inequivalent ETFs, or the triple products all belong to a cyclotomic field. This observation compelled Gene to formulate the following problem:

**Kopp’s Whisky Prize. **Find and such that

(i) ,

(ii) is an isolated point in , and

(iii) there exists such that .

To clarify, (i) prevents from being (the Naimark complement of) a SIC, (ii) prevents from belonging to a continuum of inequivalent ETFs, and (iii) prevents from having all triple products belonging to a cyclotomic field. (The field is the union of all cyclotomic fields.) Given a candidate solution , then (i) is easy to certify and (iii) reduces to a straightforward MAGMA query, while (ii) requires a bit more effort. One way to certify (ii) is to estimate the rank of the Jacobian matrix of the defining equations at . Complexifying these equations produces an algebraic variety of “quasi-ETFs.” If an ETF is an isolated point in , then it is also an isolated point in . For reference, Gene implemented this idea in a Mathematica notebook to certify that a particular member of satisfies (ii).

Finally, why is it called a whisky prize? The first person to discover any such and send it to Gene will be rewarded with a bottle of Dalwhinnie Winter’s Frost Single Malt Scotch Whisky, House Stark Game of Thrones Limited Edition. Mind the double-pun/homage: (House Stark Stark units) and (Game of Thrones Game of Sloanes).

Send your solution to Gene Kopp at this page.

]]>The paper writing has found a second wind between Philip Gibbs, Aubrey de Grey, Jaan Parts and Tom Sirgedas. For reference, I wanted to compile a list of related publications that have emerged since starting our project. (Feel free to any references I missed in the comments.) There has certainly been a bit of activity since Aubrey’s paper first hit the arXiv two years ago!

P. Ágoston, Probabilistic formulation of the Hadwiger–Nelson problem.

F. Bock, Epsilon-colorings of strips, Acta Math. Univ. Comenianae (2019) 88: 469-473.

J. Parts. A small 6-chromatic two-distance graph in the plane, Geombinatorics, vol. 29, No. 3 (2020), pp.111-115.

J. Parts. Graph minimization, focusing on the example of 5-chromatic unit-distance graphs in the plane, Geombinatorics, vol. 29, No. 4 (2020), pp.137-166.

]]>At this point, much of the effort is transitioning to the writing stage, which is taking place on Overleaf. See this comment to obtain writing privileges for the paper. This thread can be used to discuss the write-up as well as any remaining research items.

]]>P. Delsarte, J. M. Goethals, J. J. Seidel,

Geometriae Dedicata 6 (1977) 363–388.

I found Hans’ notes to be particularly helpful, so I’m posting them here with his permission. I’ve lightly edited his notes for formatting and hyperlinks.

Without further ado:

**— Spherical codes —**

A * spherical code* is a finite set of unit vectors in Euclidean space . Fundamental problems in discrete geometry and communication theory are concerned with the interplay between the size and the set of inner products For instance, the

More generally, we call a spherical code an * -code* when . Delsarte, Goethals and Seidel obtain upper bounds for the size of an arbitrary -code in terms of polynomials that interact nicely with the set . To see the utility of such a strategy, consider their absolute bound:

**Theorem 1** (Theorem 4.8 in [DGS77])**.** If is an -code with ,

*Proof:* Observe that the polynomial vanishes on and has degree . For each , define the polynomial function by . For each , notice , and so each is linearly independent. Moreover, each resides in the vector space of real-valued functions on that can be represented by a polynomial of degree at most . It follows that is at most the dimension of this vector space, which is given by ; see Section 2.2 here.

Theorem 1 is clearly sharp for by considering the vertices of a regular simplex. However, already for , equality in (1) is only known to occur for , where the corresponding spherical code is constructed from a set of equiangular lines in . In what follows, we derive improved upper bounds on for arbitrary -codes that depend on more detailed information about .

**— Gegenbauer polynomials —**

The linear programming bound for spherical codes in is stated in terms of * Gegenbauer polynomials* . These can be described recursively by , , and, for ,

Note the dependence on the ambient dimension . This recursion has the benefit of being concrete and the drawback of being completely unmotivated. Before we see how these polynomials are useful, we give some motivation as to why they naturally appear in the context of spherical codes. The rest of this section is based loosely on the excellent lecture notes by Vallentin.

We want an upper bound on , where is an arbitrary -code. Equivalently, we want an upper bound on the clique number of the infinite graph with vertices and an edge between exactly when . One such influential bound for finite graphs is given by the Lovasz theta number, which is defined as a semidefinite program. This was strengthened by Schrijver and subsequently extended to infinite graphs by Bachoc, Nebe, Oliveira and Vallentin. To state a specialized version of their bound, let denote the set of continuous functions , which we call * kernels*. A kernel is called

**Proposition 2.** If is an -code, then

*Proof:* Let be feasible for (3). Since , we have

and so

Rearranging yields as desired.

In Proposition 2, we may without loss of generality restrict our attention to * -invariant* kernels , where for all . In particular, if is -invariant, then for some continuous function . The Peter–Weyl theorem provides the decomposition , where is the vector space of restrictions of homogenous degree- harmonic polynomials to . Hence, we may express -invariant kernels in terms of the orthogonal projections . Set and let be a real orthonormal basis for . A kernel representation for is given by

The upshot is that every positive -invariant kernel can be expressed as with , with each kernel defined by the addition formula . Moreover, each kernel can be expressed as for a polynomial of degree , and the orthogonality of the spaces leads to the orthogonality relation

This determines the polynomials recursively, and rescaling each appropriately yields the Gegenbauer polynomials described by (2). The choice of scaling is irrelevant for our applications.

**— The linear programming bound for spherical codes —**

While we could derive the linear programming bound from Proposition 2, we instead give a concrete proof in the spirit of Delsarte, Goethals, and Seidel.

**Theorem 3** (Theorem 4.3 in [DGS77])**.** If is an -code, then

Equality in (4) occurs if and only if for all and

*Proof:* Let be feasible for (4). The key idea is to consider bounding from above and below. To begin, expand

For , is a positive kernel, and so . For an upper bound, the constraint for all provides

All together, we have with equality exactly when claimed.

Observe that the only property of the Gegenbauer polynomials that we used for Theorem 3 was that, for each , for all finite . This is weaker than each being a positive kernel, and Pfender obtained slight improvements based on this observation.

The case of equality in (4) motivates the following definition. A * -design* is a spherical code with for all . Equivalently, for every polynomial of degree at most ,

see Section 9.6 here. These highly uniform sets are good candidates for -codes of maximal cardinality. Indeed, if is a -design with and is a polynomial that is feasible for (4) that vanishes on , then every -code has size at most .

This strategy gives the exact values for the kissing numbers and . For , the 240 shortest vectors of the lattice have inner products . Hence, is a -code and . Delsarte, Goethals and Seidel showed that is a -design, and later Levenshtein and Odlyzko and Sloane independently showed that

satisfies with and . Applying Theorem 3 proves . A similar strategy with the Leech lattice yields .

Delsarte, Goethals and Seidel again use Gegenbauer polynomials to give a linear programming lower bound (Theorem 5.10 in [DGS77]) on the size of an arbitrary -design . In some sense, this lower bound is dual to Theorem 3 and the proof is similar. They give an upper bound on for spherical -designs with fixed and show that, in all cases, (Theorem 6.6 in [DGS77]).

The linear programming method has been generalized beyond Theorem 3. Musin developed a nonconvex extension to prove . Cohn and Elkies extended the linear programming method to noncompact settings, leading to the resolution of the sphere packing problems in by Viazovska and by Cohn, Kumar, Miller, Radchenko, and Viazovska. De Laat and Vallentin identified a general semidefinite programming hierarchy for problems in discrete geometry, the lowest level of which is the linear programming method. For more on the development of these methods, the reader is encouraged to consult the notes of Vallentin and Cohn.

]]>J. Jasper, E. J. King, D. G. Mixon, Game of Sloanes: Best known packings in complex projective space

If you have a packing that improves upon the current leader board, you can submit your packing to the following email address:

asongofvectorsandangles [at] gmail [dot] com

In this competition, you can **win money** if you find a new packing that achieves equality in the Welch bound; see this paper for a survey of these so-called equiangular tight frames (ETFs).

**The Jasper Prize.** John Jasper will pay US$100 to the first person who submits an ETF of vectors in such that

(i) , , and furthermore

(ii) no ETF of vectors in is currently known to exist.

To be explicit, there are 22 pairs in the “Game of Sloanes range” for which an ETF is currently known to exist:

Note: Fickus and Jasper conjecture that ETFs exist for both and ; see Conjecture 5.1 in this paper. Can you find either of these ETFs?

The goal of this competition is to find nice packings that generate new conjectures for the community to eventually prove (akin to Neil Sloane’s table of packings in real Grassmannian spaces). As an example, when building up the baseline for this competition, we happened upon a particularly nice packing of 5 points in which is the subject of the following conjecture:

**Conjecture.** The columns of the following matrix span optimally packed points in :

(This is Conjecture 8 in the paper.) If you need motivation to prove this conjecture, consider the following:

**King’s Coffee Prize.** Emily King will buy a coffee for the first person to prove the above conjecture.

(FYI – Emily is a coffee snob, so the coffee you get from her will be worth, like, two or three normal coffees.)

We look forward to your contributions to the Game of Sloanes!

]]>The purpose of this blog entry is to discuss recent observations regarding the Paley graph.

**— New bound on the clique number —**

There are several proofs that . In 2006, Maistrelli and Penman proved , and in 2013, Bachoc, Matolcsi and Ruzsa proved that holds for most . This contrasts with the best-known lower bound of . Considering this disparity between upper and lower bounds, I was excited to see progress on the upper bound this year:

**Theorem **(Hanson–Petridis 2019)**.**

The proof uses Stepanov’s method of auxiliary polynomials: Given such that , Hanson and Petridis construct a polynomial of degree with coefficients in such that every member of is a root of of multiplicity at least . It follows that .

Considering when is an even power of an odd prime, the proof necessarily makes use of the fact that is prime. We highlight how in the following proof sketch:

Fix , , and , and let denote the unique solution to the linear system

(Indeed, this Vandermonde system is invertible since the ‘s are distinct.) Define

By our choice of ‘s, an application of the binomial theorem reveals that has degree . (Here, it’s important that is a prime rather than a prime power, since otherwise the coefficient of might be zero.) Since , we have for every . This identity allows us to relate to the polynomial

which equals 1 by the binomial theorem and our choice of ‘s. In particular, for every , it holds that

To show that each is a root of with multiplicity at least , we take derivatives of before evaluating at . For example,

(Notice that division by might not be possible if were a prime power rather than a prime.) Similarly, the th derivative of vanishes on for each . This gives the desired result.

**— 3-point bounds on are computationally fast —**

Recall that the Lovasz theta number provides a semidefinite relaxation of the clique number of the complement graph. To see this, let denote the indicator vector of any vertex subset of a graph . Defining

then one may write

One may relax this integer program by recording convex constraints that satisfies:

This is known as the Lovasz theta number of , and by virtue of the relaxation, it holds that . When Lovasz first introduced the theta function, he proved that

with equality when is vertex-transitive. Since is both vertex-transitive and self-complementary, it follows that .

For a sharper bound on the clique number, one might consider a higher-order member of an appropriate SDP hierarchy. For example, Gvozdenovic, Laurent and Vallentin proposed such a hierarchy in 2009, and they evaluated its performance on the Paley graph. In this setting, it has become popular to refer to the classical Lovasz program as a 2-point bound, while higher-order programs produce -point bounds for . As an application, Gvozdenovic, Laurent and Vallentin computed 3-point bounds for , denoting them by ; see their Table 2 for explicit estimates. Unfortunately, going up the SDP hierarchy comes at a substantial computational price. Case in point, it took 4.5 hours to compute on a 3GHz processor with 1GB of RAM. In joint work with Mark Magsino and Hans Parshall, we managed to exploit symmetries in the Paley graph to reduce this runtime to under 20 seconds. Thanks to this speedup, we were able to run extensive experiments that suggest that is always away from the Hanson–Petridis bound:

Above, blue dots correspond to the 3-point bound, whereas the red curve gives the closed-form Hanson–Petridis bound. In particular, the 3-point bound improves upon the Hanson–Petridis bound in what appears to be a fraction of primes.

An easy interpretation of the 3-point bound is as follows: Since is vertex-transitive, we know that resides in a maximum clique. As such, we can restrict our search to the local graph , i.e., the subgraph of induced by the neighborhood of . In particular, it holds that

Next, the vertices of form a cyclic multiplicative group, and is circulant over this group. Since is circulant, the Lovasz SDP is invariant under conjugation by circulant permutations, and so we may average over these orbits to reduce to a linear program (this last move is common practice). While our numerical experiments solved this linear program for all primes less than 3000, we suspect that it’s possible to compute with in the millions by exploiting a fast Fourier transform in the LP solver. It would be interesting if one could construct feasible points for the dual LP that improve upon the Hanson–Petridis bound infinitely often. Our experiments suggest that the improvement would be an additive constant of 1 infinitely often, much like Bachoc, Matolcsi and Ruzsa’s improvement over the “trivial” bound.

**— Random Paley subgraphs satisfy a Kesten–McKay law —**

Select , and consider the random graph obtained by deleting vertices of independently with probability . What can be said about the statistics of such a random graph?

Intuitively, when is small, this random graph resembles the Erdos–Renyi graph with edge probability 1/2. The adjacency matrix of the latter graph is given by

where is symmetric with zeros on the diagonal and independent Rademacher variables on the off-diagonal. Since is a Wigner matrix, its spectrum follows a semi-circle law, and so enjoys a similar spectrum with an additional spike corresponding to the all-ones contribution. To corroborate our intuition, consider the following histograms; one illustrates the bulk of the spectrum of a 998-vertex instance of with and , and the other corresponds to an instance of the Erdos–Renyi graph on 998 vertices with edge probability 1/2. The largest eigenvalue of both graphs is off the chart at about 498. (Can you tell which is which?)

When is larger, we expect to inherit more of the structure from its parent graph , and therefore lose the semicircular behavior. For example, the following is a histogram of the bulk of the spectrum of an instance of with and :

In general, it turns out that the limiting spectral distribution for (where we fix and send ) is the so-called Kesten–McKay distribution. In joint work with Mark Magsino and Hans Parshall, we prove this using the moment method.

This result corresponds to a recent observation by Haikin, Zamir and Gavish: If is a random submatrix of a equiangular tight frame, then the eigenvalues of appear to exhibit a Wachter distribution with parameters determined by and . This observation will likely have implications for compressed sensing, where it’s important to have control over the conditioning of submatrices. To see the connection with the Paley graph, recall that the Paley equiangular tight frame is a real matrix with such that

where is the Seidel adjacency matrix of the Paley graph ; that is, , if , and otherwise . Furthermore, in this special case where , the Wachter distribution reduces to the Kesten–McKay distribution.

We’re excited to tackle more of the Haikin–Zamir–Gavish observations in the coming months.

]]>