I spent six weeks in February (2016) in New Haven. It was very nice to get back to Yale after more than two years. Here is a picture from the spectacular Yale’s art museum.

A few years ago I wrote about amazing developments in low dimensional topology. Ian Agol proved the Thurston’s virtual Haken conjecture. The proof relies on earlier major works by Dani Wise and others (see this post) and an important ingredient was Micha Sageev work on cubical complexes arising from 3-manifolds. I remember thinking that as a mathematician in another field it is unrealistic for me to want to understand these developments in any detail (which is not an obstacle for writing about them), but that one day I want to understand Sageev’s construction. At Yale, Ian Agol gave three lectures on his proof, mentioned a combinatorial form of Sageev construction, and gave me some references. A few days after landing Dani Wise talked at our seminar and he explained to me at lunch how it goes in details. (BTW, a few days before landing Micha Sageev gave a colloquium at HUJI that I missed.) I have some notes, and it is not so difficult so stay tuned for some details in a future post!. (**Update:** I will need some refreshing for writing about it. But I can assure you that Micha Sageev’s construction is a beautiful combinatorial construction that we can understand and perhaps use.)

Hee Oh talked at Yale about various analogs of the prime number theorem (PNT) and in some cases even of weak forms of the Riemann hypothesis for hyperbolic manifolds and for rational maps. This is based on a recent exciting paper by Oh and Dale Winter. Some results for hyperbolic manifolds were known before but moving to dynamics of rational maps is completely new and it follows a “dictionary” between hyperbolic manifolds and rational maps offered by Dennis Sullivan. It was nice to see in Oh’s lecture unexpected connections between the mathematical objects studied by three Yale mathematicians, Mandelbrot, Margulis, and Minsky. Are these results related to the real Riemann Hypothesis? I don’t know.

Going well over my head I want to tell you about somethings I learned from two lectures. It is about Weil’s conjectures for curves including the “Riemann hypothesis for curves over finite fields.” One lecture is by Peter Sarnak and the other by Ravi Vakil. Both lectures were given twice (perhaps a little differently) in Jerusalem and at Yale few weeks apart. Peter Sarnak mentioned in a talk the very concrete proof by Stepanov to the Rieman hypothesis for curves over finite fields. The proof uses some sort of the polynomial method, and it is this concrete proof that is useful for a recent work on Markoff triples by Bourgain, Gamburd and Sarnak. Ravi Vakhil mentioned a very very abstract form of the conjecture or, more precisely, of the “rationality” part of it proved in 1960 by Bernard Dwork (yes, Cynthia’s father!). It started from an extremely abstract version offered by Kapranov in 2000 to which Larsen and Lunts found a counterexample. A certain weaker form of Kapranov’s conjecture that Ravi discussed might still be correct. (A related paper to Ravi’s talk is Discriminants in the Grothendieck ring by Ravi Vakil and Melanie Matchett Wood.) These very abstract forms of the conjectues are also extremely appealing and it is especially appealing to see the wide spectrum from the very concrete to the very abstract. It is also nice that both the polynomial method and the quest for rationality (of power series) are present also in combinatorics.

Alex Lubotzky and I gave a 6-weeks course on high dimensional expanders. This was the fourth time we gave a course on the subject and quite a lot have happened since we first taught a similar course some years ago so it was quite interesting to get back to the subject. I certainly plan to devote a few posts to HD-expanders and Ramanujan complexes at some point in the future.

**Update:** There will be a special semester on high-dimensional combinatorics and the Israeli Institute for Advanced Studies in Jerusalem in the academic year 2017/2018.

At the Simons Center in NYC Rafał Latała gave a beautiful lecture on the solution by Thomas Royen of the Gaussian correlation conjecture. Here is a review paper by Rafał Latała and Dariusz Matlak.

**Update:** Here is an article about this proof in Quanta magazine.

I also gave three lectures at Yale about polymath projects (at that time both Polymath10 and Polymath11 were active), and a special welcome lecture to fresh Ph. D. potential students POLYMATH and more – Mathematics over the Internet (click for the presentation) about polymath projects and mainly Polymath5, MathOverflow, mathematical aspects of Angry Birds and why they should all choose Yale.

One of repeated rather unpleasant dreams I had over the years (less so in the last decade) was that the Israeli army discovered that I still owe some months of service, and I find myself confusingly and inconveniently back in uniform. A few weeks (+ one year) ago I had a new variant of that dream appropriately scaled to my current age: MIT discovered that I still owe some months of my postdoctoral service. This was much more pleasant!

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Oded Goldreich’s 60 birthday meeting, April 19-20 at the Weitzmann Institute promises to be a great event. Here is the webpage of the event. Many cool talks with cool titles. To celebrate the birthday we run a poll for the coolest title among eight selected titles. Please participate!

The coolest titles are:

First place: 31% of votes (22/71)

Shafi Goldwasser, Wednesday, April 19, 14:20-14:50

Second place: 20% (14/71)

Boaz Barak, Wednesday, April 19, 14:50-15:30

Third place: 17% (12/71)

Avi Wigderson, Thursday, April 20, 10:00-10:40

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European Commission will launch €1 billion quantum technologies flagship; The full quantum manifesto;

A general article about quantum technologies and computing in the Economist.

“Nature” article Quantum computers ready to leap out of the lab in 2017. It is mainly on Google, Microsoft, and Monroe’s lab mentioning also IBM, Rigetti and Quantum Circuits (a new startup by the Yale group). A little debate between John Martinis and Robert Schoelkopf about the target of “quantum supremacy.” Regarding topological quantum computing Leo Kouwenhoven declares that “2017 is the year of braiding,”

How to say “Quantum computers” in other languages? Hebrew: מחשבים קוונטים pronounced: maghshevim kvantim; Spanish: Computadoras cuánticas (google translate)*. *

Please contribute! For earlier efforts in this spirit see “more or less in various languages” and “When it rains it pours” in various languages. Of course, ordinary comments are welcome as well.

The Race to Sell True Quantum Computers Begins Before They Really Exist, (Wired) Mainly on Google and IBM. IBM Inches Ahead of Google in Race for Quantum Computing Power, MIT Technology Review. Commercialize *quantum* technologies in five years (Nature)

IBM: IBM’s quantum cloud computer goes commercial (Nature) IBM moves from a 5-qubit quantum computer to 50-qubit commercial quantum computer in the near future. Also IBM’s first commercial quantum computer paves way to overhaul of molecular simulations (chemistry world); IBM Building First Universal Quantum Computers for Business and Science (IBM Press Release); and a PC World article.

Google: Researchers Report Milestone in Developing Quantum Computer (NYT); Quantum computing is poised to transform our lives. Meet the man leading Google’s charge (Wired, and interview with John Martinis)

Microsoft: Microsoft Makes Bet Quantum Computing Is Next Breakthrough (NYT) ; Inside Microsoft’s quest for a topological quantum computer (Nature) Microsoft doubles down on quantum computing bet. Microsoft approach is based on topological quantum computing.

Others: Scientists are close to building a quantum computer that can beat a conventional one, Science, Chris Monroe and the Startup ionQ, and various other groups and methods. Quantum Computing on Cusp, EE Times, Yale’s group and Quantum Circuits. Quantum hanky-panky (Seth Lloyd, Edge); Its much bigger than it looks ( ;

NIST: National Institute of Standards and Technology, quantum divisions. Super *quantum* simulator ‘entangles’ hundreds of ions (Science daily) *Quantum computers* may have higher ‘speed limits’ than thought (Phys.org.) A thought experiment by Stephen Jordan.

D-wave. D-wave is building large scale quantum computers with (rather noisy) superconducting qubits, and implement specific optimization algorithms. *D-Wave* quantum computers: The smart person’s guide . Quantum computer learns to ‘see’ trees Science Magazine. An article in Quanta Magazine.

List of companies involved in quantum computers. A few webpages: 1Qbit ; D-wave ; Quantum circuits (Yale group) ; Rigetti ; Monroe’s blog; Station Q (Microsoft); Google; IBM-Q;

The key for large scale universal quantum computers is quantum error-correcting codes.

IBM to develop hardware to wipe out errors in quantum computing (New Scientists)

Error fix for long-lived qubits brings quantum computers nearer (New Scientist)

Physicists show that real-time *error correction* in *quantum* … (Phys.Org, reporting on research in South Africa)

New *Yale*-developed device lengthens the life of quantum information (Yale news)

A convincing demonstration of computational complexity supremacy is expected by researchers in the near future. Main approaches are using quantum circuits with around 50 qubits or via demonstration of BosonSampling with 20-30 bosons.

( NextBigFuture. )

Revealed: Google’s plan for quantum computer supremacy (New Scientist); How Widely Should We Draw The Circle? (Scott Aaronson on imminent demonstration of quantum supremacy); News release from Bristol U. ; Proto *quantum* computer inspired by Victorians gets a speed boost About BosonSampling. (New Scientist.); Massive Disruption Is Coming With Quantum Computing, Singularity Hub

Forging a Qubit to Rule Them All (Quanta Magazine); Inside the Knotty World of ‘Anyon’ Particles (Quanta Magazine); Why Insights of Nobel Physicists Could Revolutionize 21st-Century (Scientific Computing): Via Majorana fermions: Physicists In China Detect The *Majorana Fermion* (Asian Scientist Magazine); The discovery of *Majorana fermion* (EurekAlert);

Quantum computer makes first high-energy physics simulation, Nature, about the Innsbruck group.

A general article: The trouble with quantum computing by Ben Skuse, Engineering and Technology. The author expresses overall an optimistic bottom line, but explains the difficulties and also brings my point of view.

Here is the ending paragraph:

Much like outcomes in the quantum world itself, the future of quantum computing involves a number of unknowns, none of which we can predict. When and whether quantum supremacy and universal quantum computers can be achieved is currently up in the air, but the work of Chow, Lucas, Montanaro and even Kalai is contributing to tilting the odds in its favour. And each ‘step closer’ is only that. The road to quantum computing stretches off into the distance and may have no end.

Quantum computers will allow breaking most of current cryptosystems. This leads to much effort for “post-quantum cryptography,” namely developing cryptographic methods immune to quantum attacks.

HACKING, CRYPTOGRAPHY, AND THE COUNTDOWN TO QUANTUM COMPUTING (New Yorker)

Post-Quantum Cryptography: NIST’s Plan for the Future (U.S. National Institute of Standards and Technology (NIST)) *NIST* requests ideas for crypto that can survive *quantum* computers (The Register)

An article regarding RSA conference. (bank info security.); An article in Quanta Magazine

EXCLUSIVE – Mathematically provable communication security and other applications of Quantum Key Distribution (OpenGovAsia); Developing a quantum key system to make mobile transactions safer (New Atlas); China’s 2000-km Quantum Link Is Almost Complete (IEEE Spectrum); SK Telecom, Nokia team on quantum cryptography (Telecom Asia); Chinese Physicists Achieve Record-Breaking Quantum Cryptography Breakthrough (Hecked).

Atomic clocks: *Physicists* Find That as Clocks Get More Precise, Time Gets More, ScienceAlert; JILA atomic clock mimics long-sought synthetic magnetic state; Time crystals: *Time Crystals* Are Real, But That Doesn’t Mean Time Is Crystallized (Forbes); Entanglement and theoretical physics: The quantum source of space-time (Nature); The equation that could transform physics: Researchers say ER=EPR … (Daily mail). Quantum biology: Can *Quantum Physics* Explain Consciousness? (The Atlantic) Are we ready for quantum biology?, New Scientist; A New Spin on the Quantum Brain, Quanta Magazine; quantum effects in biological systems (a conference at IIAS).

U.S. report : ADVANCING QUANTUM INFORMATION SCIENCE: NATIONAL CHALLENGES AND OPPORTUNITIES

Europe’s leap into the *quantum* computing arms race (Financial Times)

U.K. The future is quantum: solution to the world’s critical problems; Quantum computers will solve problems that conventional computers never could. (Financial times)

China: Chinese satellite is one giant step for the quantum internet (Nature); 5 reasons why *China* will rule tech, 2017 edition (computerworld. quantum computers is one of the items). *China’s* push to become a tech superpower triggers alarms abroad (Financial times, quantum computing emphasized ); *China* can help drive global progress in *quantum computing* (ZD net)

Singapore: CEO conversations: *Quantum computing*, leadership, and … (Enterprise Innovation).

Australia: Sydney University’s David Reilly part of Microsoft’s billion-dollar push, The Sydney Morning Heral; Quantum computing is the next space race says Telstras Hugh Bradlow (Financial Review, Australia).

Canada: Canada leads in the race to create *Quantum* Valley, The Globe and Mail; Canada’s PM on quantum computing.

Israel: The future is here: the quantum computer revolution (in Hebrew); Calcalist article (Hebrew); Israeli scientists discovery towards QC (in Hebrew Haaretz) ; Quantum computing breakthrough: Israeli scientists invent cannon for entangled photon clusters (International Business Times).

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Art Duval, Bennet Goeckner, Carly Klivans, and Jeremy Martin found a counter example to the Garsia-Stanley partitioning conjecture for Cohen-Macaulay complexes. (We mentioned the conjecture here.) Congratulations Art, Bennet, Carly and Jeremy! Art, Carly, and Jeremy also wrote an article on the the Partitionability Conjecture in the Notices of the AMS. (Here is an article about it in the Brown Daily Herald.) Let me tell you about the conjecture and related questions. Here are the last few sentences of the paper.

Even though statements like the Partitionability Conjecture can seem too beautiful to be false, we should remember to keep our minds open about the mathematical unknown—the reality might be quite different, with its own unexpected beauty.

I will start with an analogy.

Let *P(n)* be the family of all subsets of *{1,2,…, n}. *A family of subsets of *{1,2,…, n} is an antichain *if no set in the family contains another set in the family*. *Sperner’s theorem (see this post) asserts that for every antichain of subsets of we have .

So we have a combinatorial notion: **an antichain**, and we have a numerical consequence: **Sperner’s theorem**. Our aim is to give a structure theorem which implies the numerical consequence. A saturated symmetric chain is a chain of sets of sizes , for some .

**Symmetric chain decomposition:** *P(n)* can be partitioned into saturated symmetric chains

The partition of *P(n)* into saturated symmetric chain gives you a combinatorial structure that implies the “numerical” content of Sperner’s theorem. The Garsia-Stanley’s conjecture has a somewhat similar flavor.

Let me note that there are stronger numerical results about antichains such as the LYM inequality (see the same post) and an interesting question (I think) is:

Question: Are there decomposition theorems for *P(n)* that support stronger numerical results about antichains, such as the LYM inequality?

The general framework of our discussion is described in the following diagram:

We will consider classes of objects (usually simplicial complexes) described by algebraic or topological properties (TOP) and a smaller class defined by a combinatorial notion (RIGHT). The topological/algebraic properties has some interesting numerical consequences (LEFT). Our task is to find general decomposition theorems that are implied by the topological notions and suffice to imply the numerical consequences.

The algebraic topological notion: **acyclic simplicial complex**.

The combinatorial notion: **collapsible simplicial complexes**.

Numerical relations: (1) Euler-poincare relation, (2) Morse inequalities, (3) a certain system of non linear inequalities.

The decomposition theorem was proved by Stanley.

Fixing a field of coefficients, a simplicial complex is acyclic if all its (reduced) homology groups vanish. A simplicial complex is *collapsible* if you can reduce it to the void complex by repeated deletion of pairs of faces* (F,G)* where *G* is a facet (maximal face) *F* is contained in *G* and is not contained in any other facet and . Note that a collapsing of gives a perfect matching among the faces of (We discussed collapsible and acyclic complexes in this post, and gave Bing’s example for non-collapsible contractible complex is this follow up post.)

Let be the number of -faces of , and let . The Euler-Poincare relation asserts that , Morse inequalities (a baby version) assert that all the s are nonnegative, and (I showed that) they also satisfy some non linear relations of Kruskal-Katona nature. In the paper A combinatorial decomposition of acyclic simplicial complexes, Stanley proved that if *K* is acyclic, there is a perfect matching in the set of faces of which resembles the matching obtained by collapsing, Namely (1) , , and (2) the ‘s form a simplicial complex. The existence of this perfect matching implies the numerical consequences for acyclic complexes. (But it *does not* imply that the complex is acyclic.) Earlier, I proved that a perfect matching with the first property exists. This implies the linear (Morse) inequalities.

Zeeman conjectured that if a 2-dimensional space is contractible then is collapsible. ( is the unit interval. This innocent-looking conjecture is widely believed to be false (perhaps even outrageous) because it implies both the Poincare conjecture (now theorem) and also the Andrews-Curtis conjecture.

The algebraic topological notion: **simplicial complex with prescribed Betti-numbers**.

The combinatorial notion: **Discrete** **Morse theory**. (This is an important notion developed by Robin Forman in the mid 90s.)

Numerical relations (Bjorner and me, late 80s): (1) Euler-poincare relation, (2) Morse inequalities, (3) a certain system of non linear inequalities.

The decomposition theorem was proved by Art Duval in the paper A Combinatorial Decomposition of Simplicial Complexes.

In this case Duval proved that the set of faces in the complex admits a matching (leaving isolated faces that correspond to the Betti numbers) which implies the numerical consequences found by Bjorner and me. It resembles the structure of discrete Morse theory.

The algebraic topological notion: **Cohen-Macaulay simplicial complex**.

The combinatorial notion: **shellable simplicial complexes**.

Numerical relations: (1) The *h*-numbers are non negative , (3) a certain non linear inequalities (The *h*-vector form an *M*-vector).

A decomposition conjecture: the the Garsia Stanley conjecture (now disproved). (A weaker form was proved by Duval and Zhang.)

Richard Stanley in 1979 and Adriano Garsia in 1980 conjectured that the faces of every -dimensional Cohen-Macaulay simplicial complex can partitioned into intervals [] where is a facet (top dimensional) for every . The paper by Duval, Geockner, Klivans, and Martin, A non-partitionable Cohen-Macaulay simplicial complex, presents a counter example. A weaker form of the conjecture was proved by Duval and Zhang.

We discussed face rings, and the Cohen-Macauly property in various previous posts like the one by Eran Nevo on on the commutative algebra connection to the g-conjecture, and the happy birthday post for Richard Stanley. A -dimensional simplicial complex is Cohen-Macaulay (with respect to a field of coefficients,) if when and for every face latex when . This property is equivalent (by a deep theorem of Riesner) to the Cohen-Macaulay property for the face ring of . The ring theoretic definition of Cohan-Macaulayness can be seen as an algebraic form (on the level of vector spaces) of the combinatorial decomposition conjectured by Garsia and Stanley.

The Cohen-Macauly property implies that a certain linear combinations of face numbers called the *h*-numbers are non-negative, and in addition satisfies certain nonlinear relations originated in the 1927 work of Macaulay. For complexes that admit the partitioning conjectured by Garsia and Stanley the* h*-numbers count the number of intervals in the partition according to the dimensions of the s. (This implies the nonnegativity of the *h*-numbers.)

A pure -dimensional simplicial complex is *shellable* if it can partitioned into intervals [] where is a facet (top dimensional) for every , and the union of the first intervals is a simplicial complex for every . A shellable simplicial complex is Cohen Macaulay (a direct proof of the ring-theoretic statement was was given by Kind and Kleinschmidt in 1979.) Shelling of a simplicial complex can be seen as a special form of decomposition. It is a partitioned into intervals [] where is a facet (top dimensional) for every with the additional property that the union of the first intervals in the partition is a simplicial complex for every .

Cohen-Macauly complexes are not necessarily shellable, since shellability implies stronger homotopical properties. But even these stronger homotopical properties do not suffice, Marry Ellen Rudin found an example of non-shellable simplicial ball. (See also here.)

The algebraic topological notion: **General simplicial complex.** (With prescribed shifted complex, if you wish).

The combinatorial notion:** non-pure shellability**. (This notion introduced by Michelle Wachs is also sometimes called **michellability**.)

**A decomposition conjecture** (by me) decomposition of the shifted complex into intervals can be mimicked on the original complex. (More general than the Garsia-Stanley conjecture and thus false.) This decomposition implies the system of linear equaltions satisfied by simplicial complexes with prescribed algebraic shifting. (See, this paper on algebraic shifting, and this post on algebraic shifting.)

**A decomposition theorem (by Duval and Zhang): **decomposition of the shifted complex into intervals can be mimicked by a decomposition of the original complex into certain trees in the paper Iterated Homology and Decompositions of Simplicial Complexes.

Boolean trees

1) Find a version of Duval-Zhang decomposition theorem for Cohen-Macauly complexes that supports also the *non-linear *inequalities for the -vector. (Like the decomposition theorems of Stanley and of Duval.) Extend to the general case of Duval-Zhang theorem!

2) Is there an analog for discrete Morse theory (using either Boolean trees or full Boolean intervals) for triangulated manifolds (and more generally Buchsbaum compexes) for the usual homology and for various classes of iterated homology?

3) For the class of triangulated spheres is there a tree-decomposition a la Duval and Zhang that supports the non negativity of *g*-numbers or the full *g*-conjecture?

4) Can these partitioning theorems be extended to more general classes of regular CW complexes?

And let me end with two more sentences from the ending paragraph of the Notices paper.

The story of the Partitionability Conjecture has many facets. Shellability, partitionability, constructibility, and the Cohen-Macaulay property come from different but overlapping areas of mathematics: combinatorics, commutative algebra, topology, and discrete geometry. The hierarchy of these structural properties turned out to be more complicated than we had anticipated, just as Rudin’s and Ziegler’s examples demonstrate that even the simplest spaces can have intricate combinatorics.

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**Update: **As Lance Fortnow the chair of the selection committee commented below the deadline for this year nomination is April 2, 2017. (Nominations should be sent to Lance to fortnow@cc.gatech.edu and if you need more time nicely ask Lance). Lance wrote that the selection committee welcomes diverse applications.

Here is the description of the prize given in the prize webpage.

The Theory community benefits in many ways from the dedicated service, above and beyond the call of duty, of many of its members. Among other contributions, the field’s members underpin the operation of conferences, journals, prizes, funding agencies, and other community activities, help ensure funding for the field, and promote the recognition of the field by external communities. The SIGACT Distinguished Service Prize is intended to recognize and promote their contributions, as well as to raise awareness of the need for and importance of such service, for the health of our community.

The prize is given annually to an individual who has made substantial service contributions to the Theoretical Computer Science community. (From 2002 to 2012, the award was given biennially.)

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László Babai; Avi Wigderson; Lance Fortnow; Lane Hemaspaandra; Sampath Kannan; Hal Gabow; Richard Karp; Tom Leighton; Rockford J. Ross; Alan Selman; Michael Langston; S. Rao Kosaraju; Fred S. Roberts; Ian Parberry; David S. Johnson

**A poll: Please ignore the “answers” and enter your own answer under “other”.**

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The Ramsey number *R(s, t)* is defined to be the smallest *n* such that every graph of order *n* contains either a clique of *s* vertices or an independent set of *t* vertices. Understanding the values of *R(s,t)* is among the most important, fruitful, and frustrating questions in combinatorics.

Vigleik Angeltveit and Brendan D. McKay proved that the value of the diagonal Ramsey number * R(5,5) *is at most

The proof of is via computer verification, checking approximately two trillion separate cases! Congratulations Brendan and Vigleik! (I heard about it from Greg Kuperberg.)

The best known lower bound of *42* was established by Exoo in 1989. The previous best upper bound of 49 was proved by Brendan McKay and Stanislaw P. Radziszowski.

By this 4-days old theorem we now have *43 ≤ R(5, 5) ≤ 48*. Brendan and Vigleik write “The actual value of *R(5, 5)* is widely believed to be *43*, because a lot of computer resources have been expended in an unsuccessful attempt to construct a Ramsey *(5,5)*-graph of order *43.” *

In 1995 Brendan McKay and Stanislaw Radziszowski famously proved that **R(4, 5) = 25**. Listing all graphs with 24 vertices which admit a coloring without a clique on 5 vertices or an independent set with 4 vertices is a major part of the current proof.

Here is, more or less, how Greg described the achievement on FB: How many people can you have at a party such that no 5 are all acquainted and no 5 are all strangers? It has been known for nearly 30 years that 42 people is possible, and known for 20 years that 49 is not possible. The momentous news is that 48 is also not possible.

Greg continuation was a little cryptic to me: According to the paper, most mathematicians with any opinion on the matter agree with Douglas Adams that 42 is the answer. (By convention, the question is phrased in terms of the first impossible value, which is now known to be at least 43 and at most 48, and conjectured to be the former.) In response to my confusion Barry Simon added a ring theoretic aspect: Gil Kalai Isn’t the answer to “the ring of what” always: “one ring to rule them all” sort of like what “42” always means.

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The sign of a Latin square is the product of signs of rows (considered as permutations) and the signs of columns. A Latin square is even if its sign is 1, and odd if its sign is -1. It is easy to see that when is odd the numbers of even and odd Latin squares are the same.

**The Alon-Tarsi Conjecture:** When is even the number of even Latin square is different from the number of odd Latin square.

This conjecture was proved when and is prime by Drisko and when and is prime by Glynn. The first open case is .

A stronger conjecture that is supported by known data is that when is even there are actually** more** even Latin squares than odd Latin squares. (This table is taken from the Wolfarm mathworld page on the conjecture.)

**A poll**

Unlike previous “test your intuition” questions the answer is not known.

The Alon-Tarsi conjecture arose in the context of coloring graphs from lists. Alon and Tarsi proved a general theorem regarding coloring graphs when every vertex has a list of colors and the conjecture comes from applying the general theorem to Dinits’ conjecture that can be regarded as a statement about list coloring of the complete bipartite graph . In 1994 Galvin proved the Dinitz conjecture by direct combinatorial proof. See this post. Gian-Carlo Rota and Rosa Huang proved that the Alon-Tarsi conjecture implies the Rota basis conjecture (over ) when is even.

Let be a graph on vertices . Associate to every vertex a variable . Consider the graph polynomial Alon and Tarsi considered the coefficient of the monomial . If this coefficient is non-zero then they showed that for every lists of colors, colors for vertex , there is a legal coloring of the vertices from the lists! Alon and Tarsi went on to describe combinatorially the coefficient as the difference between numbers of even and odd Euler orientations.

Let me mention the paper by Jeannette Janssen The Dinitz problem solved for rectangles that contains a proof based on Alon-Tarsi theorem of a rectangle case of Dinitz conjecture. It is very interesting if the polynomial used by Janssen can be used to prove a “rectangular” (thus a bit weaker) version of the Rota’s basis conjecture. A similar slightly weaker form of the conclusion of Rota’s basis conjecture is conjectured by Ron Aharoni and Eli Berger in much much greater generality.

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Recently Gil asked me whether I would like to contribute to his blog and I am happy to do so. I enjoy both finite and infinite combinatorics and it seems that these fields drifted apart in recent decades. The latter is a comment not possibly substantiated by personal experience yet rather by my perception of the number and kinds of papers published. I believe that there is no mathematical reason for this separation but that it came about for reasons of the sociology of mathematics—maybe one should create a chair for this subject somewhere sometimes. Anyhow, I would like to write a few words about an area where finite and infinite combinatorics come together, the partition calculus for ordinal numbers.

The Partition Calculus, contemporarily classified as 03E02 under Mathematical Logic/Set Theory was initiated in 1956 with the article *A partition calculus in set theory,* published by Paul Erdos and Richard Rado in the Bulletin of the American Mathematical Society. asserts that for every set of -sized subsets of a set of size there is an of size such that all subsets of of size are in or an of size such that no subset of of size is in . Many mathematicians are concerned with the case where are finite. As in the finite realm the notions of cardinal number and ordinal number coincide, it is there unnecessary to differentiate between these notions of size. Furthermore, via the Well-Ordering-Theorem, the Axiom of Choice implies that for cardinalities the statement is equivalent to where denotes the smallest ordinal number of cardinality . It also implies that there is no such that . Hence, believing the Axiom of Choice, one may limit ones study to the cases in which are order-types and is a natural number. Although there are interesting open questions in contexts where the Axiom of Choice fails and also in contexts where not all of are ordinal numbers, I will for now exclude these from the discussion. That is, what I would like to talk about is the subject of transfinite Ramsey Numbers, is to say that but for no .

First I would like to discuss the lower storeys of the transfinite. It is known that generally for a countable ordinal and a natural number the Ramsey number is countable (If no subscript appears it is understood to be ). When is a finite multiple of or , the number’s calculation is similar in character to the case where is a natural number though it tends to be slightly more involved. For example where is the least number without a digraph on vertices without an independent -tuple and without an induced transitive subtournament on vertices. The are the Tournament Ramsey Numbers which have been investigated slightly more intensely in [1], [2] and exact values are known for . The last paper on these problems was published in 1997 by Jean Larson and William Mitchell in the very first issue of the Annals of Combinatorics. A degree argument yields which gives a good idea about the growth rate of as counterexamples to numbers being finite Ramsey numbers easily carry over. Furthermore they found a digraph on thirteen vertices without a transitive triple or independent quadruple thus establishing .

Recently I started to discuss these problems with Ferdinand Ihringer and Deepak Rajendraprasad. The latter found a digraph with the two aforementioned properties but on fourteen vertices and shortly thereafter they could establish by a triangle-count that there is no such digraph on vertices. So we know now that . Generally these problems should provide a nice playground for people in the business of solving Ramsey-type problems with the help of computers, cf. [3].

The situation is slightly more complicated in the case of finite multiples of as a degree argument only yields a cubic upper bound for which is a number defined such that for all natural numbers and . This is elaborated on in detail in [4].

Recently Jacob Hilton has considered problems of this kind involving additional topological structure alone (cf. [5]) and together with Andr\'{e}s Caicedo (cf. [6]).

In the next post I am going to elaborate on results and open questions

regarding Ramsey numbers for larger countable ordinals.

[1] Kenneth Brooks Reid, Jr. and Ernest Tilden Parker. Disproof of a conjecture of

Erdos and Moser on tournaments. J. Combinatorial Theory, 9 (1970).

[2] Adolfo Sanchez-Flores. On tournaments and their largest transitive subtournaments.

Graphs Combin., 10, 1994.

[3] Stanislaw P. Radziszowski. Small Ramsey numbers. Electron. J. Combin., 1:Dynamic

Survey 1, 30 pp. (electronic), 1994,

[4] Thilo Volker Weinert, Idiosynchromatic poetry. Combinatorica, 34 (2014).

[5] Andres Eduardo Caicedo and Jacob Hilton, Topological Ramsey numbers and countable ordinals.

[6] Jacob Hilton. The topological pigeonhole principle for ordinals.

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Timothy Chow launched polymath12 devoted to the Rota Basis conjecture on the polymathblog. A classic paper on the subject is the 1989 paper by Rosa Huang and Gian Carlo-Rota.

Let me mention a strong version of Rota’s conjecture (Conjecture 6 in that paper) due to Jeff Kahn that asserts that if you have bases of an -dimensional vector space then you can choose such that for every , the vectors form a basis of and also the vectors form a basis. This conjecture can be regarded as a strengthening of the Dinits Conjecture whose solution by Galvin is described in this post. Rota’s original conjecture is the case where depends only on .

A very nice special case of Rota’s conjecture was proposed by Jordan Ellenberg in the polymath12 thread: Given trees on vertices, is it always possible to order the edges of each tree so that the th edges in the trees form a tree for every ?

The February 2017 issue of the Notices of the AMS has two beautiful papers on topics we discussed here. Henry Cohn wrote an article A conceptual breakthrough in sphere packing about the breakthrough on sphere packings in 8 and 24 dimensions (see this post) and Art Duval, Carly Klivans, and Jeremy Martin wrote an article on the the Partitionability Conjecture. (That we mentioned here.)

I have some plans to write about the partitionability conjecture (and an even more general conjecture of mine) soon. But now I would like to draw your attention to a weakening of these conjectures, still implying the “numerical” consequences of the original conjectures, that was proved in 2000 by Art Duval and Ping Zhang in their paper Iterated homology and decompositions of simplicial complexes . The partitionability conjecture is about decompositions into subcubes (or intervals in the Boolean lattice) and the result is about decomposition into subtrees of the Boolean lattice. (See here for the massage “trees not cubes!” in another context.)

This pictures was taken by Edna Wigderson in a 50th birthday party for Avi Wigderson. Unfortunately the picture shows us while landing from our 3 meter high (3.28 yards) jumps and thus does not fully capture the achievement.

The guy on the left is Bernard Chazelle, a great computer scientist and geometer, a long time friend of me and Avi, and the father of Damien Chazelle the director and writer of the movie La La Land now nominated for the record 14 Oscar awards. I wish Damien to win a record number of Oscars and to continue writing, directing, and producing wonderful movies so as to keep shattering his own records and giving excitement and joy to hundreds of millions, perhaps even billions, of people. (Update: Six Oscars, Damien the youngest ever director to win.)

For those in Israel let me draw your attention to the Jerusalem Baroque orchestra and especially to the 2017 Bach Festival. (I thanks Menachem Magidor for telling me about this wonderful orchestra.)

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