A constant lower bound for the union-closed sets conjecture — mentioned by thomasahle

Tue, 03 Jan 2023 21:09:01 GMT

Paper link: https://arxiv.org/abs/2211.09055 Abstract: We show that for any union-closed family F\subseeq 2^[n],F≠{∅}, there exists an i∈[n] which is contained in a 0.01 fraction of the sets in F. This is the first known constant lower bound, and improves upon the Ω(log_2(|F|)^{−1}) bounds of Knill and Wójick. Our result follows from an information theoretic strengthening of the conjecture. Specifically, we show that if A,B are independent samples from a distribution over subsets of [n] such that Pr[i∈A]<0.01 for all i and H(A)>0, then H(A∪B)>H(A).

Paper and discussion history. A mention is not an endorsement.

A constant lower bound for the union-closed sets conjecture — mentioned by sischoel

Sat, 19 Nov 2022 10:53:35 GMT

From skimming the paper (https://arxiv.org/abs/2211.09055), the proof seems to be amazingly short and relies only on a few theorems that one encounters in introductory class on information theory. Definitely recommendable to read if one is interested in that kind of combinatorics.

Paper and discussion history. A mention is not an endorsement.

A constant lower bound for the union-closed sets conjecture · HackerPapers discussions

A constant lower bound for the union-closed sets conjecture — mentioned by thomasahle

A constant lower bound for the union-closed sets conjecture — mentioned by sischoel