The Polymath10 project on the Erdos-Rado Delta-System conjecture took place over this blog from November 2015 to May 2016. I aimed for an easy-going project that people could participate calmly aside from their main research efforts and the duration of the project was planned for one year. I also wanted to propose and develop my own homological approach to the problem.
The purpose of this post is to (belatedly) formally announce that the project has ended, to give links to the individual posts and to briefly mention some advances and some thoughts about it.
The posts were
- Polymath10: The Erdos Rado Delta System Conjecture, Posted Nov 2, 2015. (138 comments)
- Polymath10, Post 2: Homological Approach, Posted Nov 10, 2015. (125 comments.)
- Polymath 10 Post 3: How are we doing?, Posted Dec 8, 2015. (103 comments.)
- Polymath10-post 4: Back to the drawing board?, Posted Jan 31, 2016. (11 comments.)
- Polymath 10 Emergency Post 5: The Erdos-Szemeredi Sunflower Conjecture is Now Proven. Posted May 17, 2016. (35 comments.)
- Polymath 10 post 6: The Erdos-Rado sunflower conjecture, and the Turan (4,3) problem: homological approaches, Posted on May 27, 2016. (5 comments.)
The problem was not solved and we did not come near a solution. The posts contain some summary of the discussions, a few results, and some proposals by the participants. Phillip Gibbs found a remarkable relation between the general case and the balanced case. Dömötör Palvolgyi shot down quite a few conjectures I made, and Ferdinand Ihringer presented results about some Erdos-Ko-Rado extensions we considered (In term of upper bounds for sunflower-free families). Several participants have made interesting proposals for attacking the problem.
I presented in the second post a detailed homological approach, and developed it further in the later threads with the help of Eran Nevo and a few others. Then, after a major ingredient was shot down, I revised it drastically in the last post.
Participants made several computer experiments, for sunflower-free sets, for random sunflower-free sets, and also regarding the homologica/algebraic ideas.
The posts (and some comments) give some useful links to literature regarding the problem, and post 5 was devoted to a startling development which occurred separately – the solution of the Erdos-Szemeredi sunflower conjecture for sunflowers with three petals following the cup set developments. (The Erdos-Szemeredi sunflower conjecture is weaker than the Erdos-Rado conjecture.)
The origin of my homological approach
A (too) strong version of the homological conjecture appeared in my 1983 Ph. D. thesis written in Hebrew. The typesetting used the Hebrew version of Troff.