⭐ ❗ 💡 😎 😀 hi all. this blog was started with the idea of advocating/ promoting/ experimenting with cyber collaboration in mind. some of this has indeed materialized over the years (now entering 4^{th} year into this blog), some of it “has yet to.” polymath was mostly a gleam in the eye of a few mathematicians when this blog started, but its now had several very notable successes, the largest one with the Zhang twin prime bounds improvements. Nielsens awesome book on cyber collaboration was published just a few yrs before this blog inception, but it seems few have read it and its rarely cited.

recently Kalai has started a collaboration on the Erdos sunflower problem. this is a decades old problem that shows up in very deep/ cool complexity theory. found it myself almost ~1½ decade ago in monotone circuit lower bounds proofs and did allude/ write it up briefly before on the Erdos100 post.**[8]** the original proofs by Razborov did not have sunflower constructions but later very sharp/ innovative authors (mainly **[13]**?) were able to identify/ isolate/ extract/ build them out of his seminal results.

to me this is one of the real golden threads in advanced TCS & extremal set/ circuit theory, possibly quite ripe for expanding, but few have noticed it. Razborov seems not to have said much about sunflowers at all. Rossman has built on these results in his Phd thesis and related paper now about ~½ decade old.**[18]**

also just a few years ago, a seemingly breakthrough result by Alon et al ties sunflower bounds to matrix multiplication lower bounds.**[11]** thats a very big deal. matrix multiplication lower bounds are considered one of the top open problems of TCS and there are musings/ hints by “someone” it might connect to major complexity class separations such as P vs NP.**[17]**

Jukna the great circuit combinatorialist has some nice summaries of sunflower bounds in his latest outstanding book on circuit theory.

acc to Kalai apparently Tao has called for computer experiments with small sunflowers (*where?!?* it seems quite rare for elite mathematicians to talk about computer experiments, at times it seems to be *verboten* or some kind of *faux paus*… which reminds me of Bells book title *“speakable and unspeakable in QM”*…) but it appears nobody has published anything whatsoever on the topic so far, even in cyberspace. it seems quite amenable to simple computer experiments but none of the conjectures are described in terms of algorithms. have long had some strong temptation to try but would like to converse with expert collaborators online more before jumping into that to avoid neophyte glitches.

alas though, despite all the commotions in blog comments by leading lights (incl Gowers and in his own blog), so far nobody engaged elsewhere has shown up in stackexchange chat rooms despite the multiple invitations/ entreaties/ pleas. hey well “rome wasnt built in a day,” it took many years before mathematicians embraced blogs and comments, so one just has to continue to be patient eh?

spking of matrix multiplication lower bounds, have a bunch of links & intend to post on that sometime. (it seems likely to have more near-future developments given all the attn and am gonna try to time it with the next one.)

fine print/ disclaimer: have conversed/ cyber chatted at long length with the author of **[19]** and while he has a Phd, talking to other experts and analyzing his own work, fyi have not run across any “endorsement” so far by anyone… *ie, caveat emptor* 😐

**a. sunflower**

- 1. Sunflower (mathematics) – Wikipedia, the free encyclopedia
- 2. The Erdos-Rado sunflower lemma – Polymath1Wiki
- 3. Polymath10: The Erdos Rado Delta System Conjecture | Combinatorics and more
- 4. Polymath10, Post 2: Homological Approach | Combinatorics and more
- 5. Polymath 10 Post 3: How are we doing? | Combinatorics and more
- 6. Cup Sets, Sunflowers, and Matrix Multiplication | Combinatorics and more
- 7. Sunflower conjecture | Speedup in Computational Complexity
- 8. Erdös100—tribute to a brilliant contrarian | Turing Machine
- 9. sunflower conjecture | Combinatorics and more
- 10. Gil Kalai starts Polymath10 | Gowers’s Weblog
- 11. On Sunflowers and Matrix Multiplication/ Alon, Shpilka, Umans
- 12. Sunflowers/ Jukna
- 13. SYMMETRIC APPROXIMATION ARGUMENTS FOR MONOTONE LOWER BOUNDS WITHOUT SUNFLOWERS / Berg, Ulfberg
- 14. co.combinatorics – Detailed Materials on Sunflowers – Theoretical Computer Science Stack Exchange
- 15. co.combinatorics – The state of art for sunflower system – Theoretical Computer Science Stack Exchange
- 16. co.combinatorics – The state of art of the sunflower lemma – MathOverflow
- 17. np hardness – Does solving matrix multiplication in quadratic time imply that SETH is false? – Theoretical Computer Science Stack Exchange
- 18. CiteSeerX — The Monotone Complexity of k-Clique on Random Graphs/ Rossman
- 19. [1408.3671] Asymptotic Improvement of the Sunflower Bound
- 20. Boolean Function Complexity: Advances and Frontiers (Algorithms and Combinatorics, Vol. 27): Stasys Jukna: 9783642245077: Amazon.com: Books
- 21. Approximate bases, sunflowers, and nonstandard analysis | What’s new

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