i) Definitions:
Let $F$ be a union closed family (U.C. family for short)
Let $S$ := $\cup_{ \Omega \in F} \Omega $ (the support of the family).
Let $F_{x} $ denote the members of $F$ containing $x$.

A) Union Closed Average variation : ( with $S$ being $T_0$-separated by $F$)

$ \sum_{x \in S}{ |F_{x}| } is \ge |F|.|S|/2 $.
( That is average $F_x$ size (on $S$) is greater than $|F|/2$)

Note: I cannot find any counter-example.
The $T_0$ separation of $ S $ by $ F $ means that for any two different points ${x,y} $ there exist a member of $F$ containing one point and not the other(you can choose which!) .

Another avenue of generalization that I cannot dismiss is a the multiset variation.

B) Mutltiset variation :
F is a family of $(\Omega,\eta_{\Omega})$ where:

• the $\Omega$ forms an U.C family
• $\eta$ takes its values in $\mathbb {N}^{\gt 0}$ ( or $\mathbb {R^{\gt 0}}$ or $ [1,2,..,p]$ ordered naturally )
• for any $ \Omega_1 , \Omega_2$ of $F$ : $ \eta_{\Omega_1 \cup \Omega_2} \ge Sup ( \eta_{\Omega_1} ,\eta_{\Omega_2} )$

Now the conjecture is the best measured $F_x$ is at least half that of $F$ where measure is the sum of members measure : the standard U.C. conjecture being that of $\eta$ being the constant one function.

C) The general point of view: (not strictly a generalization as asked by M.O.),
The U.C. problem is I believe of a fundamental nature, it pertains to very weak structures in the following sense: in algebra the weak (or basic or atomic) structures are the ideals, in the case of an order (or lattice) it is an upset (a family stable by overclusion , otherwise said an over-set of any member is a member), whereas we have in this problem something even weaker or more basic: stable by max only. All this comes of observing first that U.C. conjecture is easy of upsets family and still unsolved otherwise and second that algebraic techniques are not very helpful.

i) Definitions:
Let $F$ be a union closed family (U.C. family for short)
Let $S$ := $\cup_{ \Omega \in F} \Omega $ (the support of the family).
Let $F_{x} $ denote the members of $F$ containing $x$.

A) Union Closed Average variation : ( with $S$ being $T_0$-separated by $F$)

$ \sum_{x \in S}{ |F_{x}| } is \ge |F|.|S|/2 $.
( That is average $F_x$ size (on $S$) is greater than $|F|/2$)

Note: I cannot find any counter-example.
The $T_0$ separation of $ S $ by $ F $ means that for any two different points ${x,y} $ there exist a member of $F$ containing one point and not the other(you can choose which!) .

Another avenue of generalization that I cannot dismiss is a the multiset variation.

B) Mutltiset variation :
F is a family of $(\Omega,\eta_{\Omega})$ where:

• the $\Omega$ forms an U.C family say $F_0$
• $\eta : F_0 \rightarrow \mathbb {N}^{\gt 0}$ ( or $\mathbb {R^{\gt 0}}$ or $ [1,2,..,p]$ ordered naturally )
• for any $ \Omega_1 , \Omega_2$ of $F_0$ : $ \eta_{\Omega_1 \cup \Omega_2} \ge Sup ( \eta_{\Omega_1} ,\eta_{\Omega_2} )$

Now the conjecture is: The best measured $F_x$ is at least half that of $F$, where measure means the sum of members measure : the standard U.C. conjecture appears when $\eta$ is the constant one function.

C) The general point of view: (not strictly a generalization as asked by M.O.),
The U.C. problem is I believe of a fundamental nature, it pertains to very weak structures in the following sense: in algebra the weak (or basic or atomic) structures are the ideals, in the case of an order (or lattice) it is an upset (a family stable by overclusion , otherwise said an over-set of any member is a member), whereas we have in this problem something even weaker or more basic: stable by max only. All this comes of observing first that U.C. conjecture is easy for upsets family and still unsolved otherwise and second that algebraic techniques are not very helpful.

i) Definitions:
Let $F$ be a union closed family (U.C. family for short)
Let $S$ := $\cup_{ \Omega \in F} \Omega $ (the support of the family).
Let $F_{x} $ denote the members of $F$ containing $x$.

**A) Union Closed Average variation ** : ( with $S$ being $T_0$-separated by $F$)

$ \sum_{x \in S}{ |F_{x}| } is \ge |F|.|S|/2 $.
( That is average $F_x$ size (on $S$) is greater than $|F|/2$)

Note: I cannot find any counter-example.
The $T_0$ separation of $ S $ by $ F $ means that for any two different points ${x,y} $ there exist a member of $F$ containing one point and not the other(you can choose which!) .

Another avenue of generalization that I cannot dismiss is a the multiset variation.

B) Mutltiset variation :
F is a family of $(\Omega,\eta_{\Omega})$ where:

• the $\Omega$ forms an U.C family
• $\eta$ takes its values in $\mathbb {N}^{\gt 0}$ ( or $\mathbb {R^{\gt 0}}$ or $ [1,2,..,p]$ ordered naturally )
• for any $ \Omega_1 , \Omega_2$ of $F$ : $ \eta_{\Omega_1 \cup \Omega_2} \ge Sup ( \eta_{\Omega_1} ,\eta_{\Omega_2} )$

Now the conjecture is the best measured $F_x$ is at least half that of $F$ where measure is the sum of members measure : the standard U.C. conjecture being that of $\eta$ being the constant one function.

i) Definitions:
Let $F$ be a union closed family (U.C. family for short)
Let $S$ := $\cup_{ \Omega \in F} \Omega $ (the support of the family).
Let $F_{x} $ denote the members of $F$ containing $x$.

A) Union Closed Average variation : ( with $S$ being $T_0$-separated by $F$)

$ \sum_{x \in S}{ |F_{x}| } is \ge |F|.|S|/2 $.
( That is average $F_x$ size (on $S$) is greater than $|F|/2$)

Note: I cannot find any counter-example.
The $T_0$ separation of $ S $ by $ F $ means that for any two different points ${x,y} $ there exist a member of $F$ containing one point and not the other(you can choose which!) .

Another avenue of generalization that I cannot dismiss is a the multiset variation.

B) Mutltiset variation :
F is a family of $(\Omega,\eta_{\Omega})$ where:

• the $\Omega$ forms an U.C family
• $\eta$ takes its values in $\mathbb {N}^{\gt 0}$ ( or $\mathbb {R^{\gt 0}}$ or $ [1,2,..,p]$ ordered naturally )
• for any $ \Omega_1 , \Omega_2$ of $F$ : $ \eta_{\Omega_1 \cup \Omega_2} \ge Sup ( \eta_{\Omega_1} ,\eta_{\Omega_2} )$

Now the conjecture is the best measured $F_x$ is at least half that of $F$ where measure is the sum of members measure : the standard U.C. conjecture being that of $\eta$ being the constant one function.

C) The general point of view: (not strictly a generalization as asked by M.O.),
The U.C. problem is I believe of a fundamental nature, it pertains to very weak structures in the following sense: in algebra the weak (or basic or atomic) structures are the ideals, in the case of an order (or lattice) it is an upset (a family stable by overclusion , otherwise said an over-set of any member is a member), whereas we have in this problem something even weaker or more basic: stable by max only. All this comes of observing first that U.C. conjecture is easy of upsets family and still unsolved otherwise and second that algebraic techniques are not very helpful.