For any set $X$ of size $n \geqslant 4$, what is the maximum cardinality of a family $\mathscr F$ of subsets of $X$ such that $\vert A \vert =4$ for all $A \in \mathscr F$, and $\vert A \cap B \vert \leqslant 1$ for all distinct $A,B \in \mathscr F$? More generally, for any $1 \leqslant m \leqslant n$, what is the maximum size of a family $\mathscr F$ of subsets of $\{1,,\ldots,n\}$ such that $\vert A \vert = m$ for all $A \in \mathscr F$, and $\vert A \cap B \vert \leqslant m$ for all distinct $A,B \in \mathscr F$?
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Do you mean $|A \cap B| \leq 1$ in the last line, instead of $\leq m$? – user1001001 Jan 07 '22 at 18:16
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I agree that some context would be helpful for this problem. Closely related: Covering with most possible equal size subsets having pairwise singleton intersections – hardmath Jan 07 '22 at 18:35
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As I think @user1001001 noticed, with its current phrasing your more general problem, in which $|A\cap B| \le m$, is easily solved by taking $\mathscr F$ to be all $\binom{n}{m}$ of the $m$-subsets of ${1,\ldots,n}$. So presumably you have something different in mind? – hardmath Jan 07 '22 at 23:02
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Yes, I meant $|A \cap B | \leqslant 1$ in the last line. Thank you. – user462634 Jan 09 '22 at 08:12
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On page number 190 (Section 13.6) of the book "Extremal Combinatorics, Second Edition" by Jukna, the following result, attributed to Frankl and Wilson, is proved:
If $L\subset \mathbb{N}$ is a finite set of integers and if $F$ is a family of subsets of an $n$ element set such that for all $A, B \in F$, $A \neq B$, $|A\cap B| \in L$, then $|F|\leq \sum_{i = 0}^{|L|}\binom{n}{i}$.
user1001001
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Applied to the first part of the Question, where $L = {0,1}$, this gives the upper bound $|\mathscr F| \le 1 + n + n(n-1)/2$. – hardmath Jan 08 '22 at 17:05
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Thank you. However, we need the additional constraint $|A|=|B|=k$ (e.g., $k=4$), which certainly reduces the cardinality of the upper bound. – user462634 Jan 09 '22 at 08:16
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@user462634 Possibly, although its not immediately obvious that the bound can be pushed down in the case when cardinalities are equal. – user1001001 Jan 09 '22 at 13:14