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.

]]>The biggest development in the previous thread:

The method used for finding this graph is vaguely described here and here. It seems that the method is currently more of an art form than an algorithm. A next step might be to automate the art away, code up any computational speedups that are available, and then throw more computing power at the problem.

]]>**Emily King** — Algebra, Geometry, and Combinatorics of Subspace Packings

**Romanos Malikiosis** — Group frames, full spark, and other topics

**John Jasper **— Equiangular tight frames from nonabelian groups

**Gene Kopp** — SIC-POVM existence and the Stark conjectures

**Matt Fickus** — Equiangular Tight Frames from Group Divisible Designs

**Mark Magsino** — A Delsarte-Style Proof of the Bukh–Cox Bound

**Joey Iverson **— Doubly transitive lines: Symmetry implies optimality

**Gary Greaves** — Equiangular lines in and the characteristic polynomial of a Seidel matrix

**Fabricio Machado** — -point semidefinite programming bounds for equiangular lines

**Marcin Pawlowski** — Using quantum information techniques to find the number of mutually unbiased bases in any given dimension

**Frederic Matter** — Detection of Ambiguities in Linear Arrays in Signal Processing

**Clayton Shonkwiler** — Symplectic Geometry and Frame Theory

**Tom Needham** — Symplectic Geometry, Optimization and Applications to Frame Theory

**Hans Parshall **— The optimal packing of eight points in

**Bill Martin** — Spherical configurations with few angles

Interest in this project has spiked since approaching (and passing) our original deadline of April 15. For this reason, I propose we extend the deadline to October 15, 2019. We can discuss this in the Polymath proposal page.

Here are some recent developments:

- The fractional chromatic number of the plane is at least 3.98.
- The smallest known 5-chromatic unit distance graph has 529 vertices and 2630 edges.
- Dömötör has an exciting idea for finding a human-verifiable proof that the chromatic number of the plane is at least 5.

I’m interested to see if this last point has legs!

]]>Activity on this project has slowed considerably, as we’ve gone 6 months without having to roll over to a new thread. As mentioned in the original thread, the deadline for this project is April 15, 2019, so we only have a couple of weeks remaining. Dömötör and Aubrey took the time to summarize the highlights of what we’ve accomplished in the last year (see below). While we don’t have a single killer result to publish, there are several branches of minor results that warrant publication. Feel free to comment on additional results that were missed in the summaries below, as well as possible venues for publication.

Dömötör Pálvölgyi:

**Finding 5-chromatic UD graph with few vertices.**The record 553-vertex graph has been published by Marijn, not much news.**Probabilistic formulation.**Here I believe we have quite interesting results, my student is planning to write his MSc thesis on it (due June).**Bichromatic origin.**Here again we have some nice small graphs (in the plane, still looking for them on the sphere), and mathematically intriguing conjectures, which would imply improved bounds for the chromatic number of the space (if we find that graph on the sphere). About an attack on a conjecture by similar copies of almost monochromatic sets with limited success (and known limitations) we will soon publish something with Nora Frankl and Tamas Hubai.**Finding bounds for the width of the largest disk/strip that can be k-colored.**We have some bounds.**Determining the chromatic number of the extended rational field Q[w1,w2,..] for certain wi’s.**I know there are some results here, but what?

Aubrey de Grey:

**Fractional CN.**Jaan Parts has improved on the published record, but no one has yet computed the FCN of a 5-chromatic graph; however I put Jaan in contact with Marijn, who has much better hardware, so that may change soon. Then the question will be to find how to get a FCN above 4; presumably it will need at least a few interlocking 5-chromatic subgraphs, and that might be a clue for a 6-chromatic strategy.**Small graphs in higher dimensions.**There has been a fair bit of interest in my 59-vertex 6-chromatic graph in R^3, which is unpublished, so it should probably be described here. For completeness it may also be worth mentioning my 7-chromatic 24-vertex one in R^4, since it’s the best that has been achieved by explicit spindling and rotating and such like. My gut feeling is that the current records for higher dimensions can be beaten by spindling and rotation if only we can come up with some kind of pan-dimensional modular toolkit with which to build larger graphs, so it may be worth laying out what is so far known as a way to motivate such work.**Upper bounds in higher dimensions.**Philip coded up a nice fast way to find the most efficient colouring of a permutahedron-based tiling in higher dimensions and provided answers for R^5 and R^6; the number is still growing at about 3^n, so there seems to be a strong case for finding better tilings, especially ones in which tiles have fewer too-near neighbours. We should probably include my tiling of the plane in which each tile has onlly 16 too-near neighbours (as against 18 for the hexagonal tiling) as a way to motivate such a search in higher dimensions.**Fraction of the plane that can be k-coloured.**We have improvements (due to Jaan) for k=5 and 6, while Croft’s solution for k=4 remains unimproved (and also solves k=1, 2, 3).**The general concept of clamping.**Three families of 5-chromatic graphs (mine, Marijn’s and Dan/Geoff’s) are constructed by identifying simpler graphs with particular properties of this or that subset of vertices in any 4-colouring, and combining them so as to preclude all such properties being simultaneously true. There is sure to be masses to discover here. Way back in the first thread we had a brief exchange about finding simple M’s at the expense of more complex L’s (using graph names from my paper) but it never really went anywhere. A promising approach could be to find hexagonally symmetric graphs in which there is a hexagon (of non-unit radius) of vertices in which all three colours different from the centre are represented in any 4-colouring; I bet there are human-provable examples of that. We already have ones in which a hexagon of radius 8/3 has to all be the same colour as the centre, and there might be human-provable versions of that too.**Measurable CNP.**Marijn found a graph with 5617 vertices for which (in his words) it was “computationally hard” to find a 5 coloring with the center vertex being bichromatic. Sounds like it’s worth keeping looking for one that can’t be done, which would give MCNP >= 6.**Siamese tilings.**It turned out to be not particularly obvious how to tile the plane with seven colours in such a way that no two points of the same colour are exactly 1 apart but yet there are, irreducibly, separate tiles less than 1 apart. We never did anything much with such tilings, but they do seem to me to be so qualitatively different from ones in which all pairs of tiles are >=1 apart that we should say something about this.**SAT-based approaches to proving the nonexistence of a 6-tiling.**Boris’s original approach couldn’t find unscaleable tilings, but my suggestion for a variation seems to be theoretically more promising; however, Boris didn’t make it work. I think we should document what we did achieve so that others have a starting-point.- On Sept 29th, Jaan optimised Philip’s brilliant 6-tiled disk from June 17th. It ended up making sense! (in terms of regular pentagons)
- Just to be sure we remember: our bounds on Euclidean dimensions of k-coloured things need to include both the largest k-tileable disks or strips and the smallest disks or strips containing a graph that is not k-colourable.