In my research the following problem appeared (and if it is true, this solves positively several my conjectures):

Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set (usually $n$ is infinite).

I call a *staroid* such an $n$-ary relation $f$ between sets (which are subsets of some fixed set $U$) such that:

- $X\notin f$ if any component $X_i$ of $X$ is empty set.
- $\{(k,I\cup J)\} \cup L\in f\Leftrightarrow \{(k,I)\} \cup L\in f \vee \{(k,J)\} \cup L\in f$ for every index $k\in n$, sets $I,J\in\mathscr{P}U$ and an indexed family $L$ of subsets f $U$ such that $\operatorname{dom} L=n\setminus\{k\}$.
- If $X\in f$ and $\forall i\in n:Y_i\supseteq X_i$ then $Y\in f$ for every $n$-indexed family $Y$ of subsets of $U$.

Let $f$ be a staroid.

Let $X$ be an $n$-indexed family of subsets of $U$ and $Y\in f$. Let also $\forall L \in f, i \in n : L_i \cap X_i \ne \varnothing$.

Conjecture: $(\lambda i\in n: X_i\cap Y_i) \in f$.

Note that $(\lambda i\in n: X_i\cap Y_i) = \{ (i,X_i\cap Y_i) | i\in n \}$ (and it is a function defined on the set $n$).