A new proof for Keevash’s theorem on the existence of designs was discovered by Stefan Glock, Daniela Kühn, Allan Lo, and Deryk Osthus! The proof is given in the paper The existence of designs via iterative absorption, and the paper contains also some new applications of the method of proof. This is great news! A second proof to a major difficult theorem is always very very important and exciting. Keevash’s theorem gave a vast generalization of the problem for decompositions of hypergraphs to complete subhypergraphs and the new theorem is even a much more general hypergraph decomposition theorem. Congratulations!

One of the important open problems about designs is the existence of q-analogs. The first example was given in 1987 by Simon Thomas. Michael Braun, Tuvi Etzion , Patric R. J. Östergard , Alexander Vardy, Alferd Wasserman found remarkable new q-designs. See also this article: Researchers found mathematical structure that was thought not to exist. Congratulations! It is an interesting question if the new existence methods apply to q-analogs (and perhaps in greater generality for all sort of algebraic gadgets).

As part of a project with Nati Linial and Yuval Peled I was interested in finding a* k*-dimensional simplicial complex on *k(k-1)* vertices with a complete *(k-1)*-dimensional skeleton, with vanishing rational homology so that every *(k-1)* face is included in the same number of *k*-faces. (This “same number” must be *k*.) Better still I want all links of i-faces to be combinatorially the same. For *k*=2 the 6-vertex triangulation of is an example, but I did not have any other example. I asked about it on MathOverflow and GNiklasch identified a remarkable example for *k=3*. (And there are some hopes for *k=4*.) Actually, I need to devote a post to MathOverflow experiences. I got answers there to several problems that intrigued me for decades.

One more thing: Daniela Kühn and Deryk Osthus were involved in recent years (sometimes with coauthors) in knocking out some very important problems in graph theory and extremal combinatorics. Their ICM14 survey describes some of their works related to Hamiltonian cycles including their solution to the famous Kelly’s conjecture.

]]>

I am quite fond of (and a bit addicted to) Nate Silver’s site FiveThirtyEight. Silver’s models tell us what is the probability that Hillary Clinton will win the elections. It is a very difficult question to understand how does the model relates to reality. What does it even mean that Clinton’s chances to win are 81.5%? One thing we can do with more certainty is to compare the predictions of one model to that of another model.

Some data from Nate Silver. Comparing the chance of winning with the chance of winning of the popular vote accounts for “aggregation of information,” the probability for a **recount** accounts for noise sensitivity. The computation for the winning probabilities themselves is also similar in spirit to the study of noise sensitivity/stability.

This data is month-old. Today, Silver gives probability above 80% for Clinton’s victory.

Given two candidates “zero” and “one” and a fixed , suppose that every voter votes for “one” with probability and for “zero” with probability and that these events are statistically independent. Asymptotically complete aggregation of information means that with high probability (for large populations) “one” will win.

Aggregation of information for the majority rule was studied by Condorcet in what is known as the “Condorcet’s Jury theorem”. The US electoral rule which is a two-tier majority with some weights also aggregates information but in a somewhat weaker way.

The data in Silver’s forecast allows to estimate aggregation of information based on actual polls which give different probabilities for voters in different states. This is reflected by the relation between the probability of winning and the probability for winning the popular vote. Silver’s data allows to see for this comparison if the the simplistic models behave in a similar way to the models based on actual polls.

We talked about Condorcet’s Jury theorem in this 2009 post on social choice.

**Marie Jean Nicolas Caritat, marquis de Condorcet (1743-1794)**

Suppose that the voters vote at random and each voter votes for each candidate with probability 1/2 (again independently). One property that we ask from a voting method is that the outcomes of the election will be robust to noise of the following kind: Flip each ballot with probability *t* for . “Noise stability” means that if *t* is small then the probability of such random errors in counting the votes to change the identity of the winner is small as well. The majority rule is noise stable and so is the US election rule (but not as much).

How relevant is noise sensitivity/stability for actual elections? One way to deal with this question is to compare noise sensitivity based on the simple model for voting and for errors to noise sensitivity for the model based on actual polls. Most relevant is Silver’s probability for “recount.”

Nate Silver computes the probability of victory for every candidate based on running many “noisy” simulations based on the outcomes of the polls. (The way different polls are analyzed and how all the polls are aggregated together to give a model for voters’ behavior is a separate interesting story.)

We talked about noise stability and elections in this 2009 post (and other places as well).

The Banzhaf power index is the probability that a voter is pivotal (namely her vote can determine the outcome of the election) based on each voter voting with equal probability to each candidate. The Shapley-Shubik power index is the probability that a voter is pivotal under a different a priory distribution for the individual voters (under which the votes are positively correlated). Nate silver computes certain power indices based on the distribution of votes in each states as described by his model. Of course, voters in swing states have more power. It could be interesting to compare the properties of the abstract power indices and the more realistic ones from FiveThirtyEight. For example, the Banzhaf power indices sum up to the square root of the size of the population, while the Shapley-Shubik power indices sum up to one. It will be interesting to check the sum of pivotality probabilities under Silver’s model. (I’d guess that Silver’s model is closer to the Shapley-Shubik behavior.)

We talked about elections, coalition forming and power measures here, here and here.

In some earlier post we considered (but *did not* recommend) the HEX election rule. FiveThirtyEight provides a tool to represent the states of the US on a HEX board where sizes of states are proportional to the number of electoral votes.

According to the HEX rule one candidates wins by forming a continuous right-left path of winning states, and the other wins by blocking every such path or, equivalently, by forming a north-south path of winning states. The Hex rule is not “neutral” (symmetric under permuting the candidates).

If we ask for winning a north-south path for red and an east-west path for blue then red wins. For a right-left blue path much attention should be given to Arizona and Kansas.

If we ask for winning a north-south path for blue and an east-west path for red then blue wins and the Reds’ best shot would be to try to gain Oregon.

Now with the recent rise of the democratic party in the polls it seems possible that we will witness two disjoint blue north-south paths (with Georgia) as well as a blue east-west path. For a percolation-interested democratically-inclined observer (like me), this would be beautiful.

One way to consider both two basic properties of the majority rule as sort of stability to errors is as follows:

a) (Information aggregation reformulated) If all voters vote for the better candidate and with some probability a ballot will be flipped, then with high probability as the number of voters grows, the better candidate still wins.

We can also consider a weak form of information aggregation where is a fixed small real number. One way to think about this property is to consider an encoding of a bit by a string on n identical copies. Decoding using the majority rule have good error-correction capabilities.

b) (Noise stability) If all voters vote at random (independently with probability 1/2 for each candidate) and with some small probability a ballot will be flipped, then with high probability (as get smaller) this will not change the winner.

The “anomaly of majority” refers to these two properties of the majority rule which in terms of the Fourier expansion of Boolean functions are in tension with each other.

It turns out that for a sequence of voting rules, information aggregation is equivalent to the property that the maximum Shapley-Shubik power of the players tends to zero. (This is a theorem I proved in 2002. The quantitative relations are weak and not optimal.) Noise stability implies a positive correlation with some weighted majority rule, and it is equivalent to approximate low-degree Fourier representation. (These are results from 1999 of Benjamini Schramm and me.) Aggregation of information when there are two candidates implies a phenomenon called indeterminacy when there are many candidates.

The anomaly of majority is important for the understanding of why classical information and computation is possible in our noisy world.

Frank Wilczek, on of the greatest physicists of our time, wrote in 2015 a paper about future physics were he (among many other interesting things) is predicting that quantum computers will be built! While somewhat unimpressed by factoring large integers, Wilczek is fascinated by the possibility that

A quantum mind could experience a superposition of “mutually contradictory” states

Now, imagine **quantum elections** where the desired outcome of the election is going to be a superposition Hilary and Donald (Or Hillary’s and Donald states of mind, if you wish.) For example **|Hillary>** PLUS **|Donald>**.

Can we have a quantum voting procedure which has both a weak form of information aggregation and noise stability? Weak form of information aggregation amounts for the ability to correct a small fraction of random errors. Noise stability amounts to decoding procedure which is based on low-degree polynomials. Such procedures are unavailable and proving that they do not exist (or disproving it) is on my “to do” list.

The fact that no such quantum mechanisms are available appears to be important for the understanding of why robust quantum information and quantum computation is not possible in our noisy world!

Quantum election and a quantum Arrow’s theorem were considered in the post “Democrat plus Republican over the square-root of two” by

One last point. I learned about Nate Silver from my friend Greg Kuperberg, and probably from his mathoverflow answer to a question about mathematics and social science. There, Greg wrote referring to the 2008 elections: “The main person at this site, Nate Silver, has hit 50 home runs in the subject of American political polling.” Indeed, in the 2008 elections Silver correctly predicted who will win in each of the 50 states of the US. This is clearly impressive but does it reflect Silver’s superior methodology? or of him being lucky? or perhaps suggests some problems with the methodology? (Or some combination of all answers?)

One piece of information that I don’t have is the probabilities Silver assigned in each state in 2008. Of course, these probabilities are not independent but based on them we can estimate the expected number of mistakes. (In the 2016 election the expected number of mistakes in state-outcomes is today above five.) Also here, because of dependencies the expected value accounts also for some substantial small probability for many errors simultaneously. Silver’s methodology allows to estimate the actual distribution of “for how many states the predicted winner will lose?” (This estimation is not given on the site.)

Now, suppose that the number of such errors is systematically lower than the predicted number of errors. If this is not due to lack, it may suggest that the probabilities for individual states are tilted to the middle. (It need not necessarily have bearing on the presidential probabilities.)

One mental experiment I am fond of asking people (usually before elections) is this: Suppose that just a minute before the votes are counted you can change the outcome of the election (say, the identity of the winner, or even the entire distribution of ballots) according to your own preferences. Let’s assume that this act will be completely secret. Nobody else will ever know it. Will you do it?

In 2008 we ran a post with a poll about it.

We can run a new poll specific to the US 2016 election.

I really like days of elections and their special atmosphere in Israel where I try never to miss them, and also in the US (I often visit the US on Novembers). I also believe in democracy as a value and as a tool. Often, I don’t like the results but usually I can feel happy for those who do like the results. (And by definition, in some sense, most people do like the outcomes.)

And here is a post about democracy in talmudic teachings.

Below the fold, my own opinion on the coming US election.

**The choice as I see it**

]]>

Live streaming for Avifest is available here. The program is here. Following the first two lectures I can witness that the technical quality of the broadcast is very good and the scientific quality of the lectures is superb. As this is posted Dick Karp have started his lecture. Go for it!!

]]>

Ladies and gentlemen, A midrasha (school) in honor of Alex Lubotzky’s 60th birthday will take place from November 6 – November 11, 2016 at the Israel Institute for Advanced Studies, the Hebrew University of Jerusalem. Don’t miss the event! And read this:

“Groups have always played a central role in the different branches of Algebra, yet their importance goes far behind the limits of Algebraic research. One of the most significant examples for this is the work of Alex Lubotzky. Over the last 35 years, Alex has developed and applied group theoretic methods to different areas of mathematics including combinatorics, computer science, geometry and number theory. His research led the way in the study of expander graphs, p-adic Lie groups, profinite groups, subgroup growth and many geometric counting problems. The 20th Midrasha, dedicated to Alex’s 60th birthday, will include lectures from leading mathematicians in these fields presenting their current work”

My friendship with Alex goes well over the last forty years, we shared exotic experiences in the Jordan River and the Amazon River, shared apartments at Yale, taught a course together 5 times and more.

]]>

My young friend and colleague Karim Adiprasito told me that he has funding for postdocs (with or without teaching) and students (with or without teaching), both at the Hebrew University of Jerusalem (HUJI) and the MPI/University Leipzig. Both places have a great combinatorics group, as well as highly active research groups in other areas and beautiful surroundings. If you are interested in or around the type of things Karim is doing – please send him an email to karim.adiprasito@mail.huji.ac.il (with an appropriate subject that reflects your intention to apply);

MPI=Max Plank Institute.

The HUJI part of the announcement above can be generalized! Like every year we do have funding for several postdoctoral positions in and around combinatorics for 1-3 years here at the Hebrew University of Jerusalem. The starting time is flexible. If you do research in combinatorics or in related areas you may enjoy our special environment (and weather, and sights), our lovely group of combinatorialists both in the mathematics and the computer science departments, and the other great research groups around.

We already had a few Ph. D. students coming from other countries before, but starting Fall 2017 we will make special effort to attract and accommodate foreign Ph D students.

Jordan Ellenberg and I are planning an informal workshop about the “polynomial method” around Christmas 2016 in Jerusalem.

]]>

]]>

Ladies and gentlemen, a workshop in Princeton in honor of Avi Wigderson’s 60th birthday is coming on October. It will take place at Princeton on October 5-8 2016 right before FOCS 2016. **Don’t miss the event !**

Attendance is free but registration is required. Also there are funds for travel support for students for which you should apply before August 1st. Saturday lectures are joint with FOCS 2016 and will be held at the FOCS hotel (location TBA).

Here is what Boaz Barak, one of the organizers, wrote about Avi and the event:

Avi is one of the most productive, generous and collaborative researchers in our community (see mosiac below of his collaborators). So, it’s not surprising that we were able to get a great lineup of speakers to what promises to be a fantastic workshop on Computer Science, Mathematics, and their interactions.

Do you want to tell a story about Avi? or about Avi’s mathematics? or about collaborating with Avi? or to send some words of congratulations to Avi? Or something like that? Or something different? Or some reflections on the CS 1980 Technion picture below? or some guesses regarding the picture next to Avi in the very first post of this blog?

Just Contribute a comment and earn a prize of 10 Israeli shekels. (Your only duty would be to remind me about it when we meet.)

Here are two cool posts from computational complexity about the old debate, looked at after twenty years. Post 1 Lance Fortnow about the debate; post 2 (Avi’s perspective).

Confirmed speakers are:

- Scott Aaronson – MIT
- Dorit Aharonov – Hebrew University of Jerusalem
- Noga Alon – Tel Aviv University
- Zeev Dvir – Princeton University
- Oded Goldreich – Weizmann Institute of Science
- Shafi Goldwasser – MIT

Weizmann Institute of Science - Russell Impagliazzo –

University of California, San Diego - Gil Kalai – Hebrew University of Jerusalem
- Richard Karp – University of California, Berkeley
- Nati Linial – Hebrew University of Jerusalem
- Richard Lipton – Georgia Institute of Technology
- László Lovász – Eötvös Loránd University
- Alex Lubotzky – Hebrew University of Jerusalem
- Silvio Micali – MIT
- Noam Nisan – Hebrew University of Jerusalem
- Toniann Pitassi – University of Toronto
- Alexander Razborov – University of Chicago
- Omer Reingold – Samsung Research America
- Michael Saks – Rutgers University
- Ronen Shaltiel – University of Haifa
- Madhu Sudan – Harvard University
- Eyal Wigderson – Hebrew University of Jerusalem

And here are the list of confirmed organizers:

- Sanjeev Arora – Princeton University
- Boaz Barak – Harvard University
- Ran Raz – Weizmann Institute of Science
- Peter Sarnak – Princeton University,

Institute for Advanced Study - Amir Shpilka – Tel Aviv University

]]>

We will not regard attacks on the sunflower conjecture based on the recent solution of the cap set problem and the Erdos Szemeredi sunflower conjecture (which is a weaker version of the Erdos-Rado conjecture that we considered in post 5) as part of the present polymath10 project, but, of course, I will be happy to hear also reports on progress or comments on these directions. Some updates on this story: Eric Naslund and Will Sawin gave a direct proof based on the polynomial method for the Erdos-Szemeredi sunflower conjecture, and an even stronger result is given by Naslund here. (Eric also has stronger quantitative bounds for Erdos-Szemeredi based on bounds for cap sets.) More cap set updates are added to this post, and can be found in the comment threads here and here.

Turan’s 1940 problem is very easy to state.

What is theminimumof edges in a 3-uniform hypergraph on n vertices with the property that every four vertices span at least one triangle?

We talked about the problem (and a little about the homological approach to it) in this post and this post.

4. 1000 dollars or so Erdos prize

If I remember correctly, the Turan’s problem is the only question not asked by Erdos himself for which he offered a monetary award.

3. The homological approach might be relevant.

This is what this post is going to be about. But while this connection is still tentative and speculative, the next connections between the two problems are solid.

2. Sasha Razborov

Sasha used the Erdos-Rado sunflower theorem in his famous theorem on superpolynomial lower bounds for monotone circuits, and his flag-algebra theory have led to remarkable progress and the best known upper bound for the Turan (4,3) problem. (See here and here .)

1. Sasha Kostochka

Sasha holds the best known upper bound for the sunflower conjecture and he found a large class of examples for the equality cases for the Turan (4,3) conjecture.

Let be a generic matrix. The -th compound matrix is the matrix of by minors. Namely, , where .

Given two -unform hypergraphs we say that and are **weakly isomorphic** if the minor of whose rows and columns correspond to sets in and respectively is non-singular. (It is fairly easy to see that if and are isomorphic then they are weakly isomorphic. This relies on the condition that is generic.) We will say that **dominates** if and is full rank. Thus, and are weakly isomorphic if each dominates the other.

Let be the -uniform hypergraph which consists of all -subsets of [n] that intersect [r]. For a -uniform hypergraph on the vertex set [n] let . For the complete -uniform hypergraph on vertices () .

We refer to a 3-uniform hypergraph with the property that every four vertices span at least one triangle as a Turan (4,3)-hypergraph.

**Conjecture: **If is a Turan (4,3)-hypergraph with vertices then

This conjecture is a refinement of Turan’s conjecture. (The conjectured 1940 lower bound by Turan can be written as .) It is known for (see this post) and an analogous statement is known for Turan (3,2)-graphs.

We would like to find even stronger algebraic statements which amounts not only to upper bounds on certain homology-like groups but to stronger statements about vanishing of certain homology-like groups.

I am also curious about

**Question:** Are all extremal examples with n vertices for the Turan (4,3) problem quasi-isomorphic? If this is the case we can also conjecture that every Turan (4,3) hypergraph dominates Turan’s example on the same number of vertices.

I hope to be able to present some experimental data on these problems.

A family of -subsets of is **shifted** if whenever and is obtained by replacing an element by a smaller element then . It can be shown that two shifted families of the same size are weakly isomorphic only if they are equal! We can use our compound matrix to describe an operation (in fact, several operations) called shifting which associated to a family a shifted family . Suppose that . is the lexicographically smallest family of sets which is weakly isomorphic to . In more details: we first consider a total ordering on -subsets of . Then we greedily build a hypergraph which is dominated by . When we reach sets we obtain a hypergraph weakly isomorphic to .

Now, if the total ordering is the lexicographic order on then we denote and call the “algebraic shifting” of . In this case, it can be shown that the resulting family is shifted.

Also of special importance is the case that is *the reverse lexicographic order*.

For sets of integers of size , the lexicographic order refers to the lexicographic order when we ordered the elements of every set from increasingly, and the reverse lexicographic order is the lexicographic order when we order the elements decreasingly.

From 12<13<14<15<…<21<22<…<31<32<… we get

and from 21<31<32<41<42<43<51<52<… we get

We mention again some connection with acyclicity and homology: is acyclic if all sets in contains ‘1’. is -acyclic iff all sets in in intersect . is -acyclic iff all sets in contains . For general , -acyclicity is not expressed by those two versions of algebraic shifting, however, is -cyclic if .

Our ultimate conjecture is:

**Main Conjecture:** If has no sunflower of size then it is -acyclic. (I.e., )

An equivalent formulation in terms of reverse lexicographic shifting is:

(**) .

Our current proposal is to use the following theorem and (a greatly modify) Conjecture B. (For general families.)

**Theorem:** Let is a family of -sets without a sunflower of size . Then

(*) For every family of -sets which is the link of a set of size (including the case , Every set in intersect .

**Conjecture B** (from post 4, greatly modified from the one discussed in posts 2,3): For a family of -sets satisfying (*) we have

(**) .

Also here, I hope to be able to present some experimental data soon.

]]>

**The Erdos-Szemeredi Sunflower Conjecture:** There is such that a family of subsets of [n] without a sunflower of size three have at most sets. (Erdos and Szemeredi have made a similar conjecture for larger sunflowers.)

**The strong Cap Set Conjecture:** There is such that a subset of without three distinct elements a, b, and c with a+b+c=0 contains at most elements.

Results by Erdos and Szemeredi give that the Erdos Rado sunflower conjecture implies the Erdos-Szemeredi sunflower conjecture. This implication is Theorem 2.3 in the paper On sunflowers and matrix multiplication by Noga Alon, Amir Shpilka, and Christopher Umans where many implications between various related conjectures are discussed (we discussed it in this post). One implication by Noga, Amir and Chris is that the ** **strong cap set Conjecture implies the Erdos-Szemeredi sunflower conjecture!

In order that the post with the cap set startling news will remain prominent, I will put the rest of this post under the fold.

We also refer the readers again to Kostochka’s review paper Extremal problem on Δ-systems, and to the paper Group-theoretic algorithms for matrix multiplication, by Henry Cohn, Robert Kleinberg, Balasz Szegedy, and Christopher Umans.

A few days ago the cap set conjecture was proved (with the upper bound , see the previous post) and this implies also the Erdos-Szemeredi sunflower conjecture! (Eric Naslund computed (via a new argument that he had found) the derived upper bound to be .)

Thus, currently the most promising line of attack to the Erdos-Rado sunflower conjecture may well be through the result and methods involved in the Erdos-Szemeredi sunflower conjecture (now theorem) via the strong cap set conjecture (now theorem). Although I don’t have much to say about it myself, discussing this avenue is certainly welcome.

Some questions that come to mind are:

- Can the polynomial method apply also to the full Erdos-Rado sunflower conjecture (for three petals).
- Is there a direct proof (via the polynomial method) for the Erdos Szemeredi conjecture
- Do the new upper bounds for the Erdos-Szemeredi conjecture have some consequences for the Erdos-Rado conjecture?
- What about the Erdos-Szemeredi conjecture for avoiding sunflowers with
*r*petals*r>3*.

In my next polymath10 post I do plan to return to the homological approach and related technology that I also like for other reasons/problems.

An off topic remark: In the previous post I expressed the (rather obvious) thought that the new cap set development may reflect also on bounds for Roth’s theorem, (little in the spirit of polymath6: “A is to B as C is to ???” ) with some naive thoughts about it. I still hope for some serious discussion about such a possibility.

For a presentation (with some nice modification – the three points on the affine line are treated symmetrically) of the Croot-Lev-Pach-Ellenberg-Gijswijt capset and further discussion see this post and comment thread on Terry Tao’s blog.

Update: Eric Naslund and Will Sawin gave a direct proof based on the polynomial method for the Erdos-Szemeredi sunflower conjecture (for three petals), and an even stronger result is given by Naslund here. I will try to give more updates on applications of the Croot-Lev-Pach-Ellenberg-Gijswijt breakthrough at the end of this post.

]]>

Briefly: it seems to me that the idea of the Croot-Lev-Pach paper I posted about yesterday

(GK: see also my last post)can indeed be used to give a new bound on the size of subsets of F_3^n with no three-term arithmetic progression! Such a set has size at most (2.756)^n. (There’s actually a closed form for the constant, I think, but I haven’t written it down yet.)Here’s the preprint. It’s very short. I’ll post this to the arXiv in a day or two, assuming I (or you) don’t find anything wrong with it, so comment if you have comments!

This is amazing! The cap set problem was quite popular here on the blog, see also Tao’s 2007 post, and Jordan made also quite an effort over the years in proving the other direction before proving this direction. (Fortunately for our profession, success for two conflicting statements was avoided.) Congratulations to Jordan, Ernie, Seva, and Peter!

Update: Congratulations also to Dion Gijswijt who also derived a similar solution to the cap set problem based on CLP! See this comment on Quomodocumque.

Updates: See also this post by Tao (presenting a symmetric version of the proof), this post by Gowers, this post in by Luca Trevisan, and this post by Peter Cameron, and this post by Anurag Bishnoi. See also this lovely quanta article Set proof stuns mathematicians by Erica Klarreich. See also the post Polynomial prestidigitation on GLL. There, among other things the relation to Smolensky’s early use of the “halving degree trick” for the polynomial method is noted. (See also this comment.)

Of course, there is also plenty of more to desire: Full affine lines for , higher dimensional affine subspaces for , some application to better bounds for Roth’s theorem, Szemeredi’s theorem, (for more, see this comment by Terry Tao,)… It is all very exciting.

Noga Alon also pointed out that the solution of the cap set problem also settles affirmatively the Erdos-Szemeredi weaker version of the Erdos-Rado Delta-system conjecture (via the connections discussed in this post) and also shows that a certain direction for showing that ω=2 for matrix multiplication cannot possibly work. The Erdos-Rado sunflower conjecture is still (at least for a few days) open.

Can the affine results be applied for integers or for combinatorial setting? The geometries are quite different but still… This is of great interest here (and also for other problems like the Kakeya problem). Starting from a positive density set in considered as a subset of we can find there a -dimensional affine subspace contained in the set. Can’t we use it (or such a subspace with a few additional pleasant properties) to get just a single combinatorial line over , or, easier, just a 3-term arithmetic progression when represent a subset of *{1,2,… ,* *}*? A bit later: These thoughts about the relevance of finite field results to questions for the integers (or reals) are not really relevant to the new discovery. But what seems to be relevant is the possibility to transfer the new method for the cap set problem back to the question on better lower bounds for Roth’s theorem.

More updates: Eric Naslund and Will Sawin gave a direct proof based on the polynomial method for the Erdos-Szemeredi sunflower conjecture, and an even stronger result is given by Naslund here. (Eric also has stronger quantitative bounds for Erdos-Szemeredi based on bounds for cap sets.) Ben Green has studied the analogue of Sarkozy’s theorem in function fields (other results on function fields are mentioned by Bloom in this comment); Variants on the CLPEG-arguments are described by Petrov and by Bishnoi over the comment threads here and here. Here is a paper by Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A. Grochow, Chris Umans, on consequences of the cap set result for fast matrix multiplication.

More updates (May 31): New applications are mentioned in a new post on quomodocumque: sumsets and sumsets of subsets including a lovely new application by Jordan, a link to a paper by Robert Kleinberg: A nearly tight upper bound on tri-colored sum-free sets in characteristic 2. And here is a link to a new manuscript by Fedor Petrov Many Zero Divisors in a Group Ring Imply Bounds on Progression–Free Subsets.

One more: (Quoting Arnab Bhattacharyya, June 15 2016 on GLL) Another amazing result that follows from these techniques for one of my favorite problems: http://arxiv.org/pdf/1606.01230v2.pdf. Fox and LM Lovasz improve the bounds for the arithmetic triangle removal lemma dramatically, from a tower of two’s to polynomial!

More (Early July 2016) : An interesting new post on Ellenberg’s blog.

]]>