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Omni-MATH / Let A be a given set with n elements. Let k<n be a given positive integer. Find the maximum value of m for which it is possible to choose sets B_(i) and C_(i) for i=1,2,…,m satisfy…

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problem

Let AA be a given set with nn elements. Let k<nk<n be a given positive integer. Find the maximum value of mm for which it is possible to choose sets BiB_i and CiC_i for i=1,2,…,mi=1,2,\ldots,m satisfying the following conditions: [list=1] [*]Bi⊂A,B_i\subset A, |Bi|=k,|B_i|=k, [*]Ci⊂BiC_i\subset B_i (there is no additional condition for the number of elements in CiC_i), and [*]Bi∩Cj≠Bj∩CiB_i\cap C_j\neq B_j\cap C_i for all i≠j.i\neq j. [/list]
Plain-text mathematical notation (without MathML)
Let A be a given set with n elements. Let k<n be a given positive integer. Find the maximum value of m for which it is possible to choose sets B_(i) and C_(i) for i=1,2,…,m satisfying the following conditions:
[list=1]
[*]B_(i)⊂A, |B_(i)|=k,
[*]C_(i)⊂B_(i) (there is no additional condition for the number of elements in C_(i)), and
[*]B_(i)∩C_(j)≠B_(j)∩C_(i) for all i≠j.
[/list]
Original LaTeX notation
Let $A$ be a given set with $n$ elements. Let $k<n$ be a given positive integer. Find the maximum value of $m$ for which it is possible to choose sets $B_i$ and $C_i$ for $i=1,2,\ldots,m$ satisfying the following conditions:
[list=1]
[*]$B_i\subset A,$ $|B_i|=k,$
[*]$C_i\subset B_i$ (there is no additional condition for the number of elements in $C_i$), and
[*]$B_i\cap C_j\neq B_j\cap C_i$ for all $i\neq j.$
[/list]

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Source and history

task_id
e52e205a-a55d-5241-9e89-2852b8539574
task_key
test--e52e205a-a55d-5241-9e89-2852b8539574
task_revision_id
1
config / split
/ test

Official source

initial import