The main purpose of this post is to start a new research thread for Polymath 10 dealing with the Erdos-Rado Sunflower problem. (Here are links to post 2 and post 1.) Here is a very quick review of where we are.
Let  be the maximum size of a family of -sets [balanced family] with no sunflower of size . Let and .
Philip Gibbs: The balanced case vs. the general case.
Phillip gave a beautiful construction showing that if is finite than . This is very interesting and I found it surprising. (Earlier we only knew that .) Of course, if you can find another construction which starts with balanced families (even only those constructed by Philip) and end with exponentially larger general families (you can even enlarge k a little,) this will provide a counterexample to the Erdos-Rado conjecture.
The homological approach:
In the previous post and some comments (like this comment and the following ones; and this comment) I continued to meditate about my homological approach. For we want to show that for balanced families, a (2m,m)-cycle will contain a sunflower with head of size smaller than m. (This will now give .) Juggeling between the homological notion of (b,c)-cycles and a combinatorial one can be useful.
Proposals by Tim:
Tim observed that if all pairwise intersections between sets in a sunflower free family have the same size then this leads to exponential upper bounds. This can be a starting point for an argument where “same size” is replaced by a weaker condition, or to ideas about how to construct a counterexample.
Random sunflower-free families
We also want to understand the expected number of sets in the sunflower-free process when we consider k-subsets fron [n]. (We would also like to understand the expected size of the union of sets in the resulting family.) This is interesting both for the general case and the balanced case. Simulations by Philip (e.g. here) and by Gonzalo (e.g. here) were presented. (I am still confused about the outcomes of the simulations, maybe we shoud run more simulations.)
In the later comments to the previous thread (starting here) that I did not digest yet, Philip offered some ideas on new constructions of various types.
Some more comments and questions
Avi Wigderson asked: Is it enough to prove the Erdos-Rado conjecture for ? ? etc? Dömötör asked the following Kneser-type coloring question:
How many colors to we need to color all k-tuples of an n element set avoiding monochromatic 3-sunflowers?
The special case mentioned by Shachar:
Shachar mentioned in comments to post I (starting here) an exciting special case (related to matrix multiplication.)
Looking at the classic papers:
It can be a good time to look at the classic papers by Abott, Hanson, and Sauer and , Spencer and Kostochka (for the general case; for sunflower of size three). Here is again the link for Kostochka’s survey. There are many other papers (even recent) about sunflowers and many aspect of the problem and related problems that we did not talk about.
Here are the six pages of Joel’s paper.