The first “official” post of this Polymath project has passed 100 comments, so I think it is time to write a second post. Again I will try to extract some of the useful information from the comments (but not all, and my choice of what to include should not be taken as some kind of judgment). A good way of organizing this post seems to be list a few more methods of construction of interesting union-closed systems that have come up since the last post — where “interesting” ideally means that the system is a counterexample to a conjecture that is not obviously false.
Standard “algebraic” constructions
If is a union-closed family on a ground set , and , then we can take the family . The map is a homomorphism (in the sense that , so it makes sense to regard as a quotient of .
If instead we take an equivalence relation on , we can define a set-system to be the set of all unions of equivalence classes that belong to .
Thus, subsets of give quotient families and quotient sets of give subfamilies.
Possibly the most obvious product construction of two families and is to make their ground sets disjoint and then to take . (This is the special case with disjoint ground sets of the construction that Tom Eccles discussed earlier.)
Note that we could define this product slightly differently by saying that it consists of all pairs with the “union” operation . This gives an algebraic system called a join semilattice, and it is isomorphic in an obvious sense to with ordinary unions. Looked at this way, it is not so obvious how one should define abundances, because does not have a ground set. Of course, we can define them via the isomorphism to but it would be nice to do so more intrinsically.
More “twisted” products
Tobias Fritz, in this comment, defines a more general “fibre bundle” construction as follows. Let be a union-closed family of sets (the “base” of the system). For each let be a union-closed family (one of the “fibres”), and let the elements of consist of pairs with . We would like to define a join operation on by
for a suitable . For that we need a bit more structure, in the form of homomorphisms whenever . These should satisfy the obvious composition rule .
With that structure in place, we can take to be , and we have something like a union-closed system. To turn it into a union-closed system one needs to find a concrete realization of this “join semilattice” as a set system with the union operation. This can be done in certain cases (see the comment thread linked to above) and quite possibly in all cases.
More specific constructions
Giving more weight to less abundant elements
First, here is a simple construction that shows that Conjecture 6 from the previous post is false. That conjecture states that if you choose a random non-empty and then a random , then the average abundance of is at least 1/2. It never seemed likely to be true, but it survived for a surprisingly long time, before the following example was discovered in a comment thread that starts here.
Let be a large integer and let be disjoint sets of size and . (Many details here are unimportant — for example, all that actually matters is that the sizes of the sets should increase fairly rapidly.) Now take the set system
To see that this is a counterexample, let us pick our random element of a random set, and then condition on the five possibilities for what that set is. I’ll do a couple of the calculations and then just state the rest. If , then its abundance is 2/3. If it is in , then its abundance is 1/2. If it is in , then the probability that it is in is , which is very small, so its abundance is very close to 1/2 (since with high probability the only three sets it belongs to are , and ). In this kind of way we get that for large enough we can make the average abundance as close as we like to
One thing I would like to do — or would like someone to do — is come up with a refinement of this conjecture that isn’t so obviously false. What this example demonstrates is that duplication shows that for the conjecture to have been true, the following apparently much stronger statement would have had to be true. For each non-empty , let be the minimum abundance of any element of . Then the average of over is at least 1/2.
How can we convert the average over into the minimum over ? The answer is simple: take the original set system and write the elements of the ground set in decreasing order of abundance. Now duplicate the first element (that is, the element with greatest abundance) once, the second element times, the third times, and so on. For very large , the effect of this is that if we choose a random element of (after the duplications have taken place) then it will have minimal abundance in .
So it seems that duplication of elements kills off this averaging argument too, but in a slightly subtler way. Could we somehow iterate this thought? For example, could we choose a random by first picking a random non-empty , then a random such that , and finally a random element ? And could we go further — e.g., picking a random chain of the form , etc., and stopping when we reach a set whose points cannot be separated further?
Complements of designs
Tobias Fritz came up with a nice strengthening that again turned out (again as expected) to be false. The thought was that it might be nice to find a “bijective” proof of FUNC. Defining to be and to be , we would prove FUNC for if we could find an injection from to .
For such an argument to qualify as a proper bijective proof, it is not enough merely to establish the existence of an injection — that follows from FUNC on mere grounds of cardinality. Rather, one should define it in a nice way somehow. That makes it natural to think about what properties such an injection might have, and a particularly natural requirement that one might think about is that it should preserve unions.
It turns out that there are set systems for which there does not exist any with a union-preserving injection from to . After several failed attempts, I found the following example. Take a not too small pair of positive integers — it looks as though works. Then take a Steiner -system — that is, a collection of sets of size 5 such that each set of size 3 is contained in exactly one set from . (Work of Peter Keevash guarantees that such a set system exists, though this case was known before his amazing result.)
The counterexample is generated by all complements of sets in , though it is more convenient just to take and prove that there is no intersection-preserving injection from to . To establish this, one first proves that any such injection would have to take sets of size to sets of size , which is basically because you need room for all the subsets of size of a set to map to distinct subsets of the image of . Once that is established, it is fairly straightforward to show that there just isn’t room to do things. The argument can be found in the comment linked to above, and the thread below it.
An example of Thomas Bloom
Thomas Bloom came up with a simpler example, which is interesting for other reasons too. His example is generated by the sets , all -subsets of , and the 6 sets , , , , , . I asked him where this set system had come from, and the answer turned out to be very interesting. He had got it by staring at an example of Renaud and Sarvate of a union-closed set system with exactly one minimal-sized set, which has size 3, such that that minimal set contains no element of abundance at least 1/2. Thomas worked out how the Renaud-Servate example had been pieced together, and used similar ideas to produce his example. Tobias Fritz then went on to show that Thomas’s construction was a special case of his fibre-bundle construction.
This post is by no means a comprehensive account of all the potentially interesting ideas from the last post. For example, Gil Kalai has an interesting slant on the conjecture that I think should be pursued further, and there are a number of interesting questions that were asked in the previous comment thread that I have not repeated here, mainly because the post has taken a long time to write and I think it is time to post it.