Talk:Union-closed sets conjecture

Survey

I don't have time right now to do this properly, but at some point we should compare our article against the survey in arXiv:1309.3297 and add whatever important parts we might be missing. —David Eppstein (talk) 05:07, 5 March 2016 (UTC)

Proof of Union- Closed Sets Conjecture by Vladimir Blinovsky∗

https://arxiv.org/pdf/1507.01270.pdf Does anybody know of a review of this paper? — Preceding unsigned comment added by Rrogers314 (talk • contribs) 21:23, 24 September 2016 (UTC)

This dissertation refutes the proof. Either way, Wikipedia does not generally consider ArXiv a reliable source. Rzvhkon (talk) — Preceding undated comment added 02:19, 30 December 2020 (UTC)

Definition of problem

Suppose the family of sets is A, B, and C with A = {*}, B = ∅, and C = ∅.

This family is clearly closed under unions, and not all the sets in the family are empty. Yet the element * is a member of less than half of the sets.

I presume that this is a misinterpretation of the problem: The sets in the family are assumed to be unequal to each other.

For this reason, I believe that this condition needs to be stated explicitly.50.205.142.50 (talk) 17:23, 5 July 2020 (UTC)

That's a family of only two sets. You've given them three names, but really there are only two of them. Things in a set (and a family of sets is itself a set) are always unequal to each other. —David Eppstein (talk) 17:24, 5 July 2020 (UTC)

As of course I am fully aware.

My point is not to use the minimal language that can be argued is technically correct.

My point is to make the article clear.

But in any case ... did you happen to notice that the article does not refer to a "set" of sets but instead to a *family* of sets. The word "family" does not have a standard definition in mathematics that would distinguish between the incorrect interpretation I posted about and the correct one.50.205.142.50 (talk) 22:51, 5 July 2020 (UTC)

The term "A family of sets" is hyperlinked to a wiki article that explicitly states that a family need not be a set and may be a multiset. As such this statement of the problem is false. Is there any justification for not changing the word 'family' to 'set', or at least stating that in this context 'family' should be interpreted as'set'. It is very confusing at present. — Preceding unsigned comment added by 79.69.184.140 (talk) 13:51, 19 February 2022 (UTC)

Incorrect Lattice Formulation

Currently, the lattice formulation section appears to misinterpret Abe (2000) in multiple ways.

This article requires the principal filter generated by some join-irreducible x to be a strict minority of the lattice (excluding Boolean lattices). However, Abe (2000) states the conjecture as "every nontrivial finite lattice L has a join-irreducible element such that the principal filter generated by the element has at most |L|/2 elements", making no mention of a strict minority for non-Boolean lattices. The confusion may stem from the later "Moreover, equality holds if and only if L is Boolean.", but this is part of the conclusion of Theorem 1.1, not the statement of the conjecture. (This may be equivalent to Conjecture 2 by Poonen (1992), which is a strengthening of the original conjecture).

The article states that Abe shows the equivalence of this formulation to the original conjecture. However, Abe states the lattice formulation without mention of the original formulation. A better source might be Poonen (1992), where Theorem 4 shows the equivalence of the original conjecture (1.) to the dual of this formulation (6.), or Enumerative Combinatorics Volume 1 by Stanley, which Poonen cites.

Also, trivial lattices with |L| < 2 should be excluded, as they have no join-irreducibles. Spacetime02 (talk) 19:16, 22 June 2024 (UTC)