Improvement on the Erdős-Kleitman conjecture via the KKL theorem
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910060106481664 |
|---|---|
| author | Ge, Gennian Wang, Jialuo Xu, Zixiang |
| author_facet | Ge, Gennian Wang, Jialuo Xu, Zixiang |
| contents | In 1974, Erdős and Kleitman conjectured that if a family $\mathcal{F}\subseteq 2^{[n]}$ contains no matching of size \(s\) and is maximal with respect to this property, then $
|\mathcal{F}|\ge \left(1-2^{-(s-1)}\right)\cdot 2^{n}. $
For decades, the best general lower bound remained the trivial $2^{n-1}$. About a decade ago, Frankl and Tokushige emphasized that obtaining a bound of the form $\left(\frac{1}{2}+\varepsilon\right)\cdot 2^n$ for some $\varepsilon>0$ is a challenging problem. A breakthrough of Bucič, Letzter, Sudakov and Tran in 2018 showed that $
|\mathcal{F}|\ge \left(1-\frac{1}{s}\right)\cdot 2^n $
via two very elegant and quite different approaches. Our main result shows that $$ |\mathcal{F}|\ge \left(
1 - \frac{1}{s + (s-2)\frac{\log n}{2\sqrt{5}n}}
\right)\cdot 2^n $$ by exploiting a connection to the cornerstone result of Kahn, Kalai and Linial on influences of Boolean functions. Independently, we can also obtain a weaker improvement combining the linear algebra method with a combinatorial twist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18948 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Improvement on the Erdős-Kleitman conjecture via the KKL theorem Ge, Gennian Wang, Jialuo Xu, Zixiang Combinatorics 05D05, 60C05 In 1974, Erdős and Kleitman conjectured that if a family $\mathcal{F}\subseteq 2^{[n]}$ contains no matching of size \(s\) and is maximal with respect to this property, then $ |\mathcal{F}|\ge \left(1-2^{-(s-1)}\right)\cdot 2^{n}. $ For decades, the best general lower bound remained the trivial $2^{n-1}$. About a decade ago, Frankl and Tokushige emphasized that obtaining a bound of the form $\left(\frac{1}{2}+\varepsilon\right)\cdot 2^n$ for some $\varepsilon>0$ is a challenging problem. A breakthrough of Bucič, Letzter, Sudakov and Tran in 2018 showed that $ |\mathcal{F}|\ge \left(1-\frac{1}{s}\right)\cdot 2^n $ via two very elegant and quite different approaches. Our main result shows that $$ |\mathcal{F}|\ge \left( 1 - \frac{1}{s + (s-2)\frac{\log n}{2\sqrt{5}n}} \right)\cdot 2^n $$ by exploiting a connection to the cornerstone result of Kahn, Kalai and Linial on influences of Boolean functions. Independently, we can also obtain a weaker improvement combining the linear algebra method with a combinatorial twist. |
| title | Improvement on the Erdős-Kleitman conjecture via the KKL theorem |
| topic | Combinatorics 05D05, 60C05 |
| url | https://arxiv.org/abs/2603.18948 |