Here, I want to show Ilan Karpas’ proof (that we mentioned in this post) that the conjecture is correct for “large” families, namely, for families of at least subsets of . Karpas’ proof is taken from his paper Two results on union closed families.

Can Fourier methods solve Frankl’s conjecture? I hope we could have a small discussion about it and related matters, and, in any case, I myself have some comments that I will leave to the comment section.

**Theorem (Karpas):** If is a union closed family of subsets of , and then there exists an element such that .

**Theorem 2 (Karpas):** If is a family of subsets of , with the property

(*) If and , then either or .

Then there exists an element such that .

The implication from Theorem 2 to Theorem 1 is very easy. If is a union-closed family and is the complement of , then satisfies (*). (Otherwise, and and their union must be in . If has at least elements then has at most elements. If , then . We note that Condition (*) is weaker than the assertion that is union closed and indeed the assertion of Theorem 2 ~~is sharp~~ do not hold when is large. (See remark below.)

For a family of subsets of its edge-boundary is defined by

The total influence of is defined by

We now define:

,

,

The influence of on A is defined by:

Note that the assertion for Frankl’s conjecture is equivalent to: For every union-closed family there is such that . Equivalently for every family whose complement is union-closed (such families are called “simply rooted”) there is such that .

Now, fix a family of subsets of , for , , if and otherwise. if and . (In other words, and .) if and . (In other words, , and .) Define also

Note that , and similarly for other types of influences.

Let be a Boolean function defined as follows: if represents a set in , and , otherwise.

We write .

It is well known and easy to see that .

Parseval’s formula gives . It follows easily from Parseval’s formula that and hence

We write .

Our assumption (*) on is that if and , then either or . In other words for every . This implies that

(1) .

We assume that Frankl’s conjecture is false namely that , for every , or equivalently that

(2) , .

Summing over all s it follows that hence

(3) .

From it follows that

(4) .

Note that

(5)

But

(6)

Combining (4,5,6) we get

, or

(7) .

(1) and (7) gives , a contradiction.

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Answers are most welcome.

Of course, understanding the asymptotic behavior of the density of densest packing of unit spheres in is a central problem in geometry. It is a long standing hope (perhaps naïve) that algebraic-geometry codes will eventually lead to examples showing that giving an exponential improvement of Minkowsky’s 1905 bound. (For more on sphere packing asymptotically and in dimensions 8 and 24 see this post.)

The result by Serge Vlăduţ from the previous post can be seen (optimistically) as a step in the direction of exponential improvement to Minkowski’s bound.

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A crucial step was a result by Ashikhmin, Barg, and Vlăduţ from 2001 showing that certain algebraic-geometry binary linear codes have exponentially many minimal weight vectors. (This result disproved a conjecture by Nati Linial and me.) This follows by applying in a very clever way early 80s constructions going back to Bos, Conway, Sloane and Barnes and Sloane, and this requires some subtle selection of the AG-codes that can be used. To quote the author: “In order to apply Constuctions D and E we need specific good curves (the curves in the Garcia Stichtenoth towers do not perfectly match our construction) and some Drinfeld modular curves perfectly suit our purposes.”

These lines of ideas and appropriate AG-codes look to me the most promising avenue towards exponential lower bound for the Borsuk problem. (See, e.g., this paper and this post) We need for that purpose AG codes **X** where the maximum weight occurs exponentially often (this need not be difficult) and that every subset of vectors that misses the maximum distance is exponentially smaller than** |X|**.

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We solve a generalised form of a conjecture of Kalai motivated by attempts to improve the bounds for Borsuk’s problem. The conjecture can be roughly understood as asking for an analogue of the Frankl-Rödl forbidden intersection theorem in which set intersections are vector-valued. We discover that the vector world is richer in surprising ways: in particular, Kalai’s conjecture is false, but we prove a corrected statement that is essentially best possible, and applies to a considerably more general setting. Our methods include the use of maximum entropy measures, VC-dimension, Dependent Random Choice and a new correlation inequality for product measures.

Actually, the original motivation for the problem was related to the cap-set problem. (Item 19 in the third post on the cap-set problem and the Frankl-Rodl’s theorem). In 2009 I wrote three posts (A, B, C) with some ideas on connections between the cap-set problem and Frankl-Wilson/Frankl-Rodl’s theorem. The first post also dicusses various ways in which Frankl-Rodl’s theorem is more general than Frankl-Wilson.

Other related posts and papers: Frankl-Wilson theorem; A 2014 paper by Keevash and Long Frankl-Rödl type theorems for codes and permutations; Posts (I, II) on the startling solution of the cap-set prolems following the works of Croot, Lev, Pach, Ellenberg, and Gijswijt.

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Here are links to earlier posts on designs. Amazing: Peter Keevash Constructed General Steiner Systems and Designs (January 2014); Séminaire N. Bourbaki – Designs Exist (after Peter Keevash) – the paper; Amazing: Stefan Glock, Daniela Kühn, Allan Lo, and Deryk Osthus give a new proof for Keevash’s Theorem. And more news on designs (Nov 2016); Midrasha Mathematicae #18: In And Around Combinatorics

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On Monday, February 19, we will have a special day on “Quantum Coding and High Dimensional Expanders” organized by Gilles Zemor.

Sandwiches will be served for lunch, alongside cut vegetables and pastries.

Our calendar: https://iiashdc.wordpress.com/events-calendar/

Place: Room 130, IIAS, Feldman Building, Givat Ram

Program:

10:00 – 11:00 Gilles Zemor. Background on quantum computing and coding.

11:30 – 12:30 Gilles Zemor. Quantum stabilizer codes and quantum LDPC codes.

14:00 – 15:00 Dorit Aharonov. Quantum locally testable codes.

15:15 – 16:15 Gilles Zemor. Quantum LDPC codes and high-dimensional expanders.

]]>For more information, here are links to my 2014 paper with Guy Kindler, my paper The Quantum Computer Puzzle on the Notices of AMS, its arxived expanded version, and my ICM2018 paper. Last but not least my cartoon post: If Quantum Computers are not Possible Why are Classical Computers Possible?

]]>We have now at HUJI a semester break, but the special semester in High dimensional combinatorics and IIAS leaded by Tali Kaufman and Alex Lubotzky is extremely active.

The Monday program continues during the semester break (Jan 26 to March 18) in the form of “special days”, each devoted to a certain topic. (On a very basic level.)

The first special day, this Monday, January 29, is organized by Anna Gundert and Uli Wagner.

The topic is pseudo-randomness. On January 30 there is an Action now day (less basic); on February 05: Stability and property testing February 12: High dimensional Ranom walks day; On February 19: Error correcting codes and high dimensional expanders; on.

I will update about the later special days later.

Place: Room 130, IIAS, Feldman Building, Givat Ram

Program:

10:00-11:00 **Dudi Mass – Random walks by decreasing differences method **

11:30-12:30 **Ron Rosenthal **– Random walks on oriented cells – part I

13:45- 14:45 **Izhar O**ppenheim – Decomposition Theorem for high dimensional random walks

15:00-16:00 **Amitay Kamber – Cutoff of random walks on hyperbolic surfaces and Ramanujan complexes**

16:30-17:30

Our calendar:

Place: Room 130, IIAS, Feldman Building, Givat Ram

Program:

10:00-11:00 Oren Becker – The basics of stability and group theoretic prerequisites

11:15-12:15 Tamar Ziegler – Polynomial testing on uniform sets

13:45-14:45 Oren Becker – Proving stability: qualitatively and quantitatively

15:00-16:00 Lev Glebsky – Ultraproducts and stability

______

Old post.

Place: Room 130, IIAS, Feldman Building, Givat Ram

I will add the programs once they will be available.

I will add the programs once they will be available.

]]>Followed by a Basic notion lecture by Frank Calegary 16:30-17:45: **The cohomology of arithmetic groups and Langlands program**

Wednesday January 24, 18:00-17:00: Akshay Venkatesh lectures on** Cryptography and the geometry of algebraic equations**. Lecture accessible also to second and third year undergraduate students.

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