**Update:** Peter Frankl brought to my attention that the very same example appeared in a paper by Dynkin and Frankl “Extremal sets of subsets satisfying conditions induced by a graph“.

### The example

As a follow up to my previous post about Gilmer’s breakthrough regarding Frankl’s conjecture, here is a very nice example (from the paper of Zachary Chase and Shachar Lovett) related to the conjecture.

Let

Consider the following families of subsets of

, and

.

Now let .

Here are some observations:

- .
- The number of pairs whose union is also in is .
- For every , as grows, no element belongs to more than sets in .

The first assertion holds because is exponentially larger (in a fractional power of ) than .

For the second assertion we need to show that a typical union of a pair of sets in belongs to . Note that the intersection of two random subsets of of size is and hence their union is of size . As it happens the solution of the equation is precisely our . So letting we get that a typical union of two sets from is in .

The third assertion follows from the fact that an element belongs to a fraction of sets in .

This shows that a natural stability version of Frankl’s conjecture is incorrect, and gives some hint on the appearance of the parameter .

### Stability result

Such a stability version is correct when we replace with . Chase and Lovett improved Gilmer’s method and proved that

**Theorem:** If is a -approximate union closed set system, where , then there is an element which is contained in a fraction of sets in , where

### An invitation for further discussion

I will be happy to see, based on Gilmer’s paper and the five follow-up papers, a detailed, as simple as possible, explanation of Frankl’s conjecture for the parameter , to learn what is involved in Sawin’s improvement that goes beyond , and to understand the counterexamples for Gilmer’s proposal towards the full conjecture, as well as thoughts, ideas and remarks of various kind on the problem.

### Links to the papers

- Justin Gilmer, A constant lower bound for the union-closed sets conjecture
- Will Sawin, An improved lower bound for the union-closed set conjecture
- Zachary Chase, Shachar Lovett, Approximate union closed conjecture
- Ryan Alweiss, Brice Huang, Mark Sellke, Improved Lower Bound for the Union-Closed Sets Conjecture
- David Ellis, Note: a counterexample to a conjecture of Gilmer which would imply the union-closed conjecture
- Luke Pebody, Extension of a Method of Gilmer
- (Dec 9.) Lei Yu, Dimension-Free Bounds for the Union-Closed Sets Conjecture

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Re: thoughts, ideas and remarks of various kind on the problem.

I can’t say I really have a solid sense of the problem yet but the last time it came up in these parts I was following a clue from Lipton & Regan toward what I saw as a differential approach. I didn’t get very far at the time but here’s a first link for what it’s worth.

Frankl, My Dear …

Looks like the definition of psi is wrong.

GK: Thanks, Imre; corrected.The example shows that the conclusion of Frankl’s conjecture does not approximately hold for approximately union-closed families in the sense that for of pairs of sets in the family the union is also in the family.

It is still possible the conclusion of Frankl’s conjecture does approximately hold for approximately union-closed families in the sense that of sets that are unions of sets in the family, belong to the family.

Indeed. If we would try to tweak Chase and Lovett’s example to this stricter approdximately union-closed definition, we would run into problems. From results about largest k-uniform families L-avoiding (the symmetric distance between any pair is not in some subset of forbidden distances L), we would only be able to take a very small fraction of the layer, if we would want to satisfy the stricter definition.

This leads me to wonder: can we find some upper bound on

,

for a union-closed family , so that each element appears in at most sets.

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You might also be interested in this note “A Useful Inequality for the Binary Entropy Function” by Ravi Boppana where he gives a simple proof of a key inequality that is at the heart of the recent progress on the union-closed sets conjecture: https://arxiv.org/abs/2301.09664. Actually he explains some interesting history on this inequality, how he used it in a quite different context over 30 years ago.