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sercej
sercej
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I am looking in the literature for references to combinatorial result of the kind of the one below. I am quite sure they (or some variations of them) should have been studied intensively, but now I am unable to find a reference. Thank you in advance for any suggestion.

"Let $X$ be a finite set and let $\mathfrak{P}$ be a finite collection of partitions of $X$. Then, under[under some hypotheses [...], there exists a partition $P$ [satisfying some restrictions, e.g., having $k$ parts...] of $X$ such that for all $p \in P$ and for all $Q \in \mathfrak{P}$ there exists $q \in Q$ such that $p \cap q = \emptyset$."

I give also a more concrete (and figurative example), hoping to make clear the kind of problems I have in mind:

"There are finitely many people. Each of them is from London, Paris, or Rome; and is (as usual) male or female. If [hypotheses] then the people can be split into two teams, so that no team has players from all the mentioned cities and of both the two sexes"

I am looking in the literature for references to combinatorial result of the kind of the one below. I am quite sure they (or some variations of them) should have been studied intensively, but now I am unable to find a reference. Thank you in advance for any suggestion.

"Let $X$ be a finite set and let $\mathfrak{P}$ be a finite collection of partitions of $X$. Then, under some hypotheses [...], there exists a partition $P$ of $X$ such that for all $p \in P$ and for all $Q \in \mathfrak{P}$ there exists $q \in Q$ such that $p \cap q = \emptyset$."

I am looking in the literature for references to combinatorial result of the kind of the one below. I am quite sure they (or some variations of them) should have been studied intensively, but now I am unable to find a reference. Thank you in advance for any suggestion.

"Let $X$ be a finite set and let $\mathfrak{P}$ be a finite collection of partitions of $X$. Then, [under some hypotheses...], there exists a partition $P$ [satisfying some restrictions, e.g., having $k$ parts...] of $X$ such that for all $p \in P$ and for all $Q \in \mathfrak{P}$ there exists $q \in Q$ such that $p \cap q = \emptyset$."

I give also a more concrete (and figurative example), hoping to make clear the kind of problems I have in mind:

"There are finitely many people. Each of them is from London, Paris, or Rome; and is (as usual) male or female. If [hypotheses] then the people can be split into two teams, so that no team has players from all the mentioned cities and of both the two sexes"

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sercej
sercej
  • 51
  • 3

Existence of a set partition satisfying some restriction

I am looking in the literature for references to combinatorial result of the kind of the one below. I am quite sure they (or some variations of them) should have been studied intensively, but now I am unable to find a reference. Thank you in advance for any suggestion.

"Let $X$ be a finite set and let $\mathfrak{P}$ be a finite collection of partitions of $X$. Then, under some hypotheses [...], there exists a partition $P$ of $X$ such that for all $p \in P$ and for all $Q \in \mathfrak{P}$ there exists $q \in Q$ such that $p \cap q = \emptyset$."