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Veronica Phan
Veronica Phan
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A conjecture relaterelated to Frankl's conjecture

Let $\mathcal{F}\subseteq2^{[n]},\emptyset\in\mathcal{F}$ be an union-closed family of sets. For $S\in\mathcal{F}$, let $w(S)$ be the number of subsets of $S$ in $\mathcal{F}$. Does there always exist real numbers $a_1,a_2,...,a_n\geq1$ such that $\prod_{i=1}^na_i=|\mathcal{F}|$ and for every $S\in\mathcal{F}$, $\prod_{i\in S}a_i\geq w(S)$.?

Motivation: Let $w_i$ be the number of sets of $\mathcal{F}$ that contain $i$. If the above conjecture true, we have $\sum_{i=1}^n\ln a_iw_i\geq\sum_{S\in\mathcal{F}}\ln w(S)$, so we just have to prove $\sum_{S\in\mathcal{F}}\ln w(S)\geq\sum_{i=1}^n\ln a_i\frac{|\mathcal{F}|}{2}=\frac{|\mathcal{F}|\ln|\mathcal{F}|}{2}$ that would imply Frankl's conjecture.

A conjecture relate to Frankl's conjecture

Let $\mathcal{F}\subseteq2^{[n]},\emptyset\in\mathcal{F}$ be an union-closed family of sets. For $S\in\mathcal{F}$, let $w(S)$ be the number of subsets of $S$ in $\mathcal{F}$. Does there always exist $a_1,a_2,...,a_n\geq1$ such that $\prod_{i=1}^na_i=|\mathcal{F}|$ and for every $S\in\mathcal{F}$, $\prod_{i\in S}a_i\geq w(S)$.

Motivation: Let $w_i$ be the number of sets of $\mathcal{F}$ that contain $i$. If the above conjecture true, we have $\sum_{i=1}^n\ln a_iw_i\geq\sum_{S\in\mathcal{F}}\ln w(S)$, so we just have to prove $\sum_{S\in\mathcal{F}}\ln w(S)\geq\sum_{i=1}^n\ln a_i\frac{|\mathcal{F}|}{2}=\frac{|\mathcal{F}|\ln|\mathcal{F}|}{2}$ that would imply Frankl's conjecture.

A conjecture related to Frankl's conjecture

Let $\mathcal{F}\subseteq2^{[n]},\emptyset\in\mathcal{F}$ be an union-closed family of sets. For $S\in\mathcal{F}$, let $w(S)$ be the number of subsets of $S$ in $\mathcal{F}$. Does there always exist real numbers $a_1,a_2,...,a_n\geq1$ such that $\prod_{i=1}^na_i=|\mathcal{F}|$ and for every $S\in\mathcal{F}$, $\prod_{i\in S}a_i\geq w(S)$?

Motivation: Let $w_i$ be the number of sets of $\mathcal{F}$ that contain $i$. If the above conjecture true, we have $\sum_{i=1}^n\ln a_iw_i\geq\sum_{S\in\mathcal{F}}\ln w(S)$, so we just have to prove $\sum_{S\in\mathcal{F}}\ln w(S)\geq\sum_{i=1}^n\ln a_i\frac{|\mathcal{F}|}{2}=\frac{|\mathcal{F}|\ln|\mathcal{F}|}{2}$ that would imply Frankl's conjecture.

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Veronica Phan
Veronica Phan
  • 1.5k
  • 8
  • 22

A conjecture relate to Frankl's conjecture

Let $\mathcal{F}\subseteq2^{[n]},\emptyset\in\mathcal{F}$ be an union-closed family of sets. For $S\in\mathcal{F}$, let $w(S)$ be the number of subsets of $S$ in $\mathcal{F}$. Does there always exist $a_1,a_2,...,a_n\geq1$ such that $\prod_{i=1}^na_i=|\mathcal{F}|$ and for every $S\in\mathcal{F}$, $\prod_{i\in S}a_i\geq w(S)$.

Motivation: Let $w_i$ be the number of sets of $\mathcal{F}$ that contain $i$. If the above conjecture true, we have $\sum_{i=1}^n\ln a_iw_i\geq\sum_{S\in\mathcal{F}}\ln w(S)$, so we just have to prove $\sum_{S\in\mathcal{F}}\ln w(S)\geq\sum_{i=1}^n\ln a_i\frac{|\mathcal{F}|}{2}=\frac{|\mathcal{F}|\ln|\mathcal{F}|}{2}$ that would imply Frankl's conjecture.