Dan Romik, see also Romik’s mathematical gallery and Romik’s moving sofa page

Sorting is one of the most important algorithmic tasks and it is closely related to much beautiful mathematics. In this post, a sorting network is a shortest path from 12…n to n…21 in the Cayley graph of S_n generated by nearest-neighbour swaps. In other words, a sorting networks is a sequence of transpositions of the form whose product is the permutation .

Sorting networks, as we just defined, appear in other contexts and under other names. Also the term “sorting networks” is sometimes used in a different way, and in particular, allows swaps that are not nearest neighbors. (See the slides for Uri Zwick’s lecture on sorting networks.)

Richard Stanley proved a remarkable formula

for the number of sorting networks or reduced decomposition of the permutation n…21. (We mentioned it in this post. )

We can also think about sorting networks as maximal chains in the weak Bruhat order of the symmetric group. Another related notion in combinatorial geometry is that of “(simple) allowable sequence of permutations”. an allowable sequence of permutation is a sequence of permutations starting with 12…n and ending with n…21 such that each permotation in the sequence is obtained by the previous one by reverseing one set or several disjoint sets of consecutive elements. See, e.g., this paper of Hagit last on two proofs of the Sylvester-Gallai theorem via allowable sequence of permutations.

Alexander Holroyd’s videotaped lecture on random sorting networks. Holroyd’s random sorting gallery (pictures below are taken from there), and his six other amazing mathematical gallerys.

Random sorting networks were considered a decade ago in a beautiful paper by Angel, Holroyd, Romik, and Virág . In this paper some brave conjectures were made

**Conjecture 1**: the trajectories of individual particles converge to random sine curves,

**Conjecture 2:** the permutation matrix at half-time converges to the projected surface measure of the 2-sphere.

There are more conjectures! I will just show you one additional picture

The rhombic tiling corresponding to a uniform random sorting network:

In two recent breakthrough papers all those conjectures were proved. The papers are

- Circular support in random sorting networks, by Dauvergne and Virag

…in which they show the tight support bound for the circle seen in the halfway permutation, as well as the tight Lipschitz bound for trajectories.

2. The Archimedean limit of random sorting networks, by Dauvergne.

…in which all the 2007 conjectures are proved. Here is the abstract:

A sorting network (also known as a reduced decomposition of the reverse permutation), is a shortest path from 12⋯n to n⋯21 in the Cayley graph of the symmetric group Sn generated by adjacent transpositions. We prove that in a uniform random n-element sorting network , that all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-t permutation matrix measures of . As a corollary of these results, we show that if is embedded into via the map τ↦(τ(1),τ(2),…τ(n)), then with high probability, the path σn is close to a great circle on a particular (n−2)-dimensional sphere in . These results prove conjectures of Angel, Holroyd, Romik, and Virag.

These papers build on previous results, including some in the 2007 paper, recent results by Gorin and Rahman from Random sorting networks: local statistics via random matrix laws, as well as those of Angel, Dauvergne, Holroyd and Virág from their paper on the local limit of random sorting networks.

Let me mention an important conjecture on sorting networks,

**Conjecture:** For every k, and the number of appearances of the transposition (k,k+1) in every sorting network is .

This is closely related to the halving line problem. The best lower bound (Klawe, Paterson, and Pippenger) behaves like . A geometric construction giving this bound is a famous theorem by Geza Toth. The best known upper bound by Tamal Dey is .

I am amazed that I did not blog about the halving lines problem and about allowable sequences of permutations in the past. (I only briefly mentioned the problem and allowable sequences of permutations in one earlier post on a certain beautiful geometric discrepancy problem.)

The permutahedron is the Cayley graph of the symmetric group S_{n} generated by the nearest-neighbour swaps (12), (23), (34) and (n-1n). (Here .) Are there analogous phenomena for the associahedron? one can ask.

(From Wikipedia )

The eight queens puzzle is the famous problem of placing eight chess queens on a chessboard so that no two queens threaten each other. The questions if this can be done and in how many different ways, as well as the extension to *n* queens on a *n* × *n* chessboard was raised already in the mid nineteen century. Apparently Gauss was interested in the problem and figured out that the number of solutions for the **8 × 8** board is 92, and Zur Luria gave a beautiful lecture about his new results on the number of solutions in our seminar earlier this week. The lecture follows Luria’s paper New bounds on the n-queens’s problem.

Before getting to Zur’s result a little more on the history taken from Wikipedia: “Chess composer Max Bezzel published the eight queens puzzle in 1848. Franz Nauck published the first solutions in 1850.^{} Nauck also extended the puzzle to the *n* queens problem, with *n* queens on a chessboard of *n* × *n* squares. Since then, many mathematicians, including Carl Friedrich Gauss, have worked on both the eight queens puzzle and its generalized *n*-queens version. In 1874, S. Gunther proposed a method using determinants to find solutions… In 1972, Edsger Dijkstra used this problem to illustrate the power of what he called structured programming. He published a highly detailed description of a depth-first backtracking algorithm.” For more on the history and the problem itself see the 2009 survey by Bell and Stevens. Let me also mention the American Mathematical Monthly paper The n-queens problem by Igor Rivin, Ilan Vardi and Paul Zimermmann. (In preparing this post I realized that there are many papers written on the problem.)

Let be the number of ways to place n non attacking queens on an *n* by *n* board, and let be the number of ways to place n non-attacking queens on an *n* by *n* toroidal board. is sequence A000170 in the On-Line Encyclopedia of Integer Sequences. The toroidal case, also referred to as the modular *n*-queens problem was asked by Polya in 1918. Polya proved that iff is 1 or 5 modulo 6. (Clearly, .)

Here are Zur Luria’s new results:

**Theorem 1:**

**Theorem 2:** , for some .

As far as I know, these are the first non-trivial upper bounds.

**Theorem 3:** For some constant , if is of the form then .

**Update:** In the lecture Zur noted that the proof actually works for all odd n such that -1 is a quadratic residue mod n.

Theorem 3 proves (for infinitly many integers) a conjecture by Rivin, Vardi, and Zimmermann, who proved exponential lower bounds. Luria’s beautiful proof can be seen (and this is how Zur Luria views it) as a simple example of the method of algebraic absorbers, and Zur mentions an earler application of a similar methods is in a paper of Potapov on counting Latin hypercubes and related objects. Rivin, Vardi and Zimmermann conjectured that the exponential generating function of has a closed form and understanding this generating function is a very interesting problem. Luria conjectures that his upper bound for is sharp.

**Luria’s conjecture:** .

In view of recent progress in constructing and counting combinatorial designs this conjecture may be within reach. Zur also conjectures that for , 3 should be replaced by some .

More on chess on this blog: Chess can be a Game of Luck, Amir Ban on Deep Junior, and quite a few other posts.

]]>A tile is a finite subset of . We can ask if can or cannot be partitioned into copies of . If can be partitioned into copies of we say that tiles .

Here is a simpe example. Let consists of 24 points of the 5 by 5 planar grid minus the center point. cannot tile .

**Test your intuition:** Does tiles for some ?

We had a poll and 58% of voters said YES. The answer is

**YES!**

As a matter of fact Adam Chalcraft have made the beautiful conjecture that every tile in tiles for some large . This conjecture was proved by Vytautas Gruslys, Imre Leader, and Ta Sheng Tan in their remarkable paper Tiling with arbitrary tiles.

**Theorem **(Vytautas Gruslys, Imre Leader, and Ta Sheng Tan): Let be a tile. Then tiles for some .

But wait, what about our tile *T?* After seeing the abstract of Imre Leader’s lecture, looking briefly at the paper which contained the 5 by 5 minus the middle example, I posted the question on my blog. But then driving to Jerusalem I suddenly was sure that there is no way in the world the hole in T can be filled up by another tile of the same shape. T is simply too fat, I thought. I must have missed something – some extra condition or subtelty that I overlooked. It turned out that my intuition was wrong already *after* I saw the right answer. (This does happen from time to time.)

When I had a chance I looked again at the paper, and saw a beautiful picture explaining how the hole can be filled in four dimension. (BTW, I don’t know what is the minimum dimension that T can tile.)

]]>In 1983 I was a postdoc in MIT and there was a lovely group of young combinatorialists around. At MIT, Noga Alon, and I, as well as Ian Goulden and a few others were postdocs, Jeff Kahn, Paul Seymour, Ira Gessel, and Anders Björner were junior faculty, Gunter Ziegler, Mark Haiman, Francesco Brenti, and Peter Shor were among the graduate students. There were many computer scientists with interest in combinatorics and at Northeastern there were two young faculty members, Marge Bayer and Dom (Dominique) de Caen working in combinatorics, and an algebraic geometer Jonathan Fine who also became interested in combinatorics.

One day, Dom de Caen saw me and told me that some months earlier he had managed to find a copy of “Convex polytopes” in a used book store and bought it for 10 dollars. He said that he knew how rare the book was and how hard it was to get it, but decided to give it to me since I would make better use of it working in this special area.

In 2003, after several years of work, a second edition was published which contained the original text and some additional material prepared by a fresh team of young researchers: Volker Kaibel, Victor Klee, and Gunter Ziegler. This was really great. I bought a copy and had the idea to send it to Dom as a form of gratitude for his previous gesture of kindness. So I tried to find his address over the Internet. I was sad to learn that Dom had passed away in 2002. You can read about Dom’s mathematics in this paper by Edwin R. Van Dam, The combinatorics of Dom de Caen.

]]>Today in Jerusalem, Leonard Schulman talked (video available!) about a recent breakthrough by Gil Cohen, Bernhard Haeupler, and Leonard Schulman in the recent paper Explicit Binary Tree Codes with Polylogarithmic Size Alphabet.

Tree codes are extremely important objects invented by Leonard Schulman in 1992-1996. You can read about it in Wigderson’s book Mathematics and Computation (see also this post) in a whole section about Error-correction of interactive communication, a theory pioneered by Leonard. Finding explicit good (linear distance, finite alphabet) tree codes is a very important problem.

(From the introduction of the new paper) “A tree code consists of a complete rooted binary tree (either infinite or of finite depth ) in which each edge is labeled by a symbol from an alphabet . There is a natural one-to-one mapping assigning each binary string to a path starting at the root, where simply indicates which child is taken in each of the steps. For a tree code, such a path naturally maps to a string over the alphabet , which is formed by concatenating the symbols along the path. This way a tree code encodes any binary string into an equally long string over . This encoding has an online characteristic because the encoding of any prefix does not depend on later symbols. In particular, any two distinct strings that agree in their first symbols also have encodings that agree in their first symbols. A tree code is said to achieve distance if the encodings of any two strings differ in at least a -fraction of the positions after their first disagreement. The rate of a tree code is . A tree code is said to be asymptotically good if it achieves both constant distance and a constant rate, namely, the alphabet size is .”

“This paper makes progress on the problem of explicitly constructing a binary tree code with constant distance and constant alphabet size.

For every constant 1 we give an explicit binary tree code with distance and alphabet size , where n is the depth of the tree. This is the first improvement over a two-decade-old construction that has an exponentially larger alphabet of size .

As part of the analysis, we prove a bound on the number of positive integer roots a real polynomial can have in terms of its sparsity with respect to the Newton basis – a result of independent interest.”

A few years ago Cris Moore and Leonard Schulman proposed an explicit constructions for good tree codes in the paper Tree codes and a conjecture on exponential sums. The construction depends on still open conjectures on new types of exponential sums.

So let me use this opportunity to advertize a paper by Leonard and me Quasi-random multilinear polynomials which was just accepted for publication in the Israel Journal of Mathematics. Unexpected cancellation is always a very interesting phenomenon that we cherish and wish to understand. For example, we expect that the sum of the Mobius function for the first *n* integers cancels almost like that of a random walk. Depending on the meaning of “almost” this is the (known) prime number theorem, the (conjectured) Riemann hypothesis, or simply false, respectively.

A little less famous example (which I personally like) is the paper by Nikola Djokic an upper bound on the sum of signs of permutations with a condition on their prefix sets solving a problem by J. Feldman, H. Knorrer, and E. Trubowitz.

Our starting point is the determinant: It is a polynomial of degree of variables and no matter what values you assign to the variables you have a massive cancellation of almost a random-walk type. There are terms and for any assignment the value is smaller than which is . We were interested in the question of finding other such polynomials.

]]>Here is a simpe example. Let consists of 24 points of the 5 by 5 planar grid minus the center point. cannot tile .

**Test your intuition:** Does tiles for some ?

If you prefer you can think about the simpler case of consisting of eight points: the 3 by 3 grid minus the center.

I forgot to add polls…

]]>**A)** Consider a finite family of unit circles. What is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors?

This is the question about the chromatic number of the plane. (Or the maximum chromatic number of planar unit-distance graphs.) By the Aubrey de Grey’s result the answer is 5, 6, or 7.

**B)** Consider a finite family of **non-overlapping** unit circles. What is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors?

Now we talk about planar unit-distance graphs where the distance between every pair of points is at least one. Those are also called Penny graphs. The answer is four.

**C)** Consider a finite family of **pairwise intersecting** unit circles. What is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors?

This is a the (finite) planar Borsuk problem. (Now we talk about planar unit-distance graphs where the distance between every pair of points is at most one.) The answer is 3, and this follows from a simple characterization these graphs.

**D)** Consider a finite family of non-overlapping circles. What is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors?

By Koebe’s circle packing theorem (see this recent post) this is precisely the four color theorem so the answer is 4.

**E)** Consider a finite family of circles. What is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors?

Unless we make some further assumption the answer is infinite. Rom Pinchasi proved that collors suffice and also gave an example of circles where the chromatic number is at least .

**F)** Consider a finite family of circles such that every point in the plane is included in at most two circles. What is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors?

Ringel’s circle conjecture (see this paper by Jackson and Ringel) asserts that the number of colors is bounded. (There is an example that five colors are required.)

**G)** Consider a finite family of pairwise intersecting circles. What is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors?

**H)** Consider a finite family of pseudocircles. Here every pseudocircle is a closed simple path and two pseudocicles are either disjoint or hace two crossing points. Two pseudocircles are adjacent if the lens described by them does not contain any point from any other pseudocircle. What is the minimum number of colors needed to color the pseudocircles so that adjacent pseudocircles are colored with different colors? (In particular, is this number finite?)

Each of the above questions give rise to a family of graphs and lead to other interesting questions about these graphs. For example, we can ask what is the maximal number of edges in each case.

For unit distance graphs (question **A**) this is the famous Erdős unit distance problem. (Here is a survey by Szemeredi).

The Maximum number of edges of Borsuk planar graphs (question **C**) is (see this proof using lice), and this immediately implies that their chromatic number is at most three.

The maximum number of edges of Penny graphs (question **B**) is known and there is an intruiging conjecture for triangle free Penny graphs.

The maximum number of edges of (simple) planar graphs (question **D**) as follows easily from Euler’s theorem.

Alon, Last, Pinchasi, and Sharir showed an upper bound of for the number of edges in the tangent graph for pairwise intersecting circles. (Question **G.**)

An intruiging conjecture by Pinchasi, Sharir, and others is that planar tangent graphs (Question **E**) with vertices can have at most edges. This conjecture proposes a profound extension of the Szemeredi–Trotter theorem. The best known upper bound is by Marcus and Tardos.

For circles that pairwise intersect, Rom Pinchasi proved a Gallai–Sylvester conjecture by Bezdek asserting that (for more than 5 circles) there is always a circle tangential to at most two other circles. This was the starting point of important studies concerning arrangement of circles and pseudo-circles in the plane.

Each of the above questions extends to higher dimensions and also here they include some well known problems. It is a famous result by Frankl and Wilson that the cromatic number of is exponential in . We now know (see this post) that there are -dimensional “hyper-penny” graphs with minimal degree that is exponential in . Is the chromatic number of such graphs can also be exponential in ?

A stress (affine stress to be precise) of a geometric graph is an assignement of weights to edges so that every vertex is in equilibrium. We can ask many of the problems above (e.g. see this post) when we assume that the graph is stress-free, namely does not admit non-trivial stresses. (When we consider circles of different sizes we probably need to consider stresses in three dimensions.) We can think about stress-freeness as saying that we have a “non coincidental” configuration.

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A major progress on an old standing beautiful problem. Aubrey de Grey proved that the chromatic number of the plane is at least 5. (I first heard about it from Alon Amit.)

The Hadwiger–Nelson problem asks for the minimum number of colors required to color the plane such that no two points at distance one from each other have the same color. The answer is referred to as the chromatic number of the plane. The problem was posed in 1950 by Edward Nelson, and related results already appeared in a paper by Hugo Hadwiger from 1945. Untill recently it was known that the answer can be 4,5,6 or 7. The Moser spindle is a simple example of a unit-distance graph with chromatic number 4, and there is a simple coloring of the plane, found by Isbell, based on the hexagonal packing with 7 colors so that no color contains a pair of points of distance 1.

Aubrey de Grey has constructed a unit distance graph with 1567 vertices which is 5-chromatic!

**Updates: **Aubrey de Grey has made now a polymath proposal over the polymath blog aimed at finding simpler constructions, namely constructions with a smaller number of vertices, or where the computerized part of the proof is simpler. Of course, this project may lead to independent verification of the result, and perhaps even insights for what is needed to replace ‘5’ with ‘6’. Noam Elkies independently proposed over MathOverflow a polymath project following Aubrey de Grey’s paper. (April 14) Dustin Mixon and Aubrey de Grey have launched Polymath16 over at Dustin’s blog. The project is devoted to the chromatic number of the plane (Wikipage) following Aubrey de Grey’s example showing that the chromatic number of the plane is at least 5.

Here is an earlier Google+ post by Terry Tao and an earlier blogpost on the new result by Jordan Ellenberg proposing to use the polynomial method to tackle the upper bound. A blog post reporting on independent verification of some of the new results is over Dustin G. Mixon’s blog Short, Fat Matrices. A post over Shtetl Optimized describes the new development along with another important development on quantum computation. Let me also mention two related old posts over Lipton and Regan’s blog (one, two). (April 19) An excellent article on Quanta Magazine by Evelyn Lamb. (April 21) A great blog post (French) on Automath.

(From Wikipedea: A seven-coloring of the plane, and a four-chromatic unit distance graph in the plane (the Moser spindle), providing upper and lower bounds for the Hadwiger–Nelson problem.) Below, two figures from Aubret de Grey’s paper.

Let me also mention the related Rosenfeld’s problem discussed in this post: Let G be the graph whose vertices are points in the plane and two vertices form an edge if their distance is an odd integer. Is the chromatic number of this graph finite?

These and related problems are discussed also in my survey article: Some old and new problems in combinatorial geometry I: Around Borsuk’s problem.

]]>David Conlon |
University of Oxford |

Michael Farber |
Queen Mary, University of London |

Howard Garland* |
Yale University |

Lev Glebsky |
Universidad Autónoma de San Luis Potosí |

Misha Gromov* |
IHES |

Venkatesan Guruswami |
Carnegie Mellon University |

Ming-Hsuan Kang |
National Chiao Tung University |

Daniela Kühn |
University of Birmingham |

James Lee |
University of Washington |

Winnie Li |
Pennsylvania State University |

Shachar Lovett |
UC San Diego |

Roy Meshulam |
Technion |

Deryk Osthus |
University of Birmingham |

János Pach |
NYU |

Peter Sarnak |
IAS, Princeton |

Leonard Schulman |
Caltech |

Uli Wagner |
IST |

Shmuel Weinberger |
University of Chicago |

Avi Wigderson |
IAS, Princeton |

Gilles Zémor |
Université de Bordeaux |

I just came back from a splendid visit to Singapore and Vietnam and I will write about it later. While I was away, Nathan Rubin organized a lovely conference on topics closed to my heart ERC Workshop: Geometric Transversals and Epsilon-Nets with many interesting lectures. Nathan himself announced a great result with new upper bounds for planar weak ε-nets. In 2007 I wrote a guest post (my first ever blog post) on the topic on Terry Tao’s blog. The next section is taken from that old post with reference to one additional result from 2008.

let be the least number such that *every *finite set X possesses at least one weak -net for X with respect to convex bodies of cardinality at most . (One can also replace the finite set X with an arbitrary probability measure; the two formulations are equivalent.) Informally, *f* is the least number of “guards” one needs to place to prevent a convex body from covering more than of any given territory.

A central problem in discrete geometry is:

**Problem**: For fixed d, what is the correct rate of growth of f as ?

It is already non-trivial (and somewhat surprising) that is even finite. This fact was first shown by Alon, Bárány, Füredi, and Kleitman (the planar case was achieved earlier by Bárány, Füredi, and Lovász), who established a bound of the form ; this was later improved to by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl. In the other direction, the best lower bound was for many years for some positive c(d); it was shown by Matousek that c(d) grows at least as fast as for some absolute constant c. In 2008 Bukh, Matousek, and Nivasch proved that .

In the plane the state of our knowledge was that (up to constants) for every ) . The new dramatic breakthrough is

**Theorem (Nathan Rubin, 2018):** .

I hope the paper will be arxived in the next few months.

Here is a page with the titles and abstracts of the lectures. A lot of very interesting advancements on Helly-type theorems, transversals, geometric-Ramsey, topological methods and other topics. Let me briefly mention one direction.

Weak -nets play an important role in the proof by Alon and Kleitman of the Hadwiger-Debrunner conjecture. The proof is so nice that I could not resist the temptation to present and discuss it in two comments (1,2) to my old 2007 posts.

Theorem (Alon and Kleitman): For every integers and , , there is a function such that the following is true:

Every finite family of convex sets in with the property that among every sets in the family there are with non empty intersection can be divided to at most intersecting subfamilies.

Dramatic improvements of the bounds for were achieved in a 2015 paper Improved bounds on the Hadwiger-Debrunner numbers by Chaya Keler, Shakhar Smorodinsky, and Gabor Tardos. This have led to further improvements and related results that Shakhar and Chaya talked about in the meeting.

There were various other developments regarding weak ε-Nets andrelated notions that occured since my 2007 post and the result of Buck, Matousek, and Nivasch in 2008, and probably I am not aware of all of them (or even forgot some). Of course, comments about related results are most welcome in the comment section. Here are links to a lecture by Noga Alon: 48 minutes on strong and weak epsilon nets (Part 1, Part 2). Let me mention just two beautiful recent developments: A paper On weak ε-nets and the Radon number by Moran and Yehudayoff gives a simple proof for their finiteness in a very abstract setting. A paper by Har-Peled and Jones How to Net a Convex Shape studies interesting weaker notions. A weaker notion of weak epsilon nets is described in the work by Bukh and Nivash in Nivash’s Ein Gedi abstract.

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