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Let $S_1,\dots,S_m \subseteq \{1,2,\dots,n\}$ be a collection of sets, each of size $k$. I'll say that they are almost-disjoint if, for every $i,j$ with $i\ne j$, we have $|S_i \cap S_j| \le 1$, i.e., each pair of sets has at most one element in common.

Is there an asymptotic upper-bound on the largest possible value of $m$, as a function of $n,k$, for any almost-disjoint collection of sets?

In particular, suppose $k=n^c$, for some $0 \le c \le 1$. Can I upper-bound $m$ as $m=O(n^{f(c)})$, for some function $f$? What's the best upper bound known?

If I replace "almost-disjoint" with "disjoint", the problem becomes an easy application of the pigeonhole principle; we find $m = O(n^{1-c})$. However, I don't know how to deal with the generalization to almost-disjoint collections. Is there some generalization of the pigeonhole principle that is applicable here? This question arose in the analysis of an algorithm I was thinking about. I wondered if it might be related to some kind of combinatorial design but I couldn't match it to any existing concept I know of.

We can use the projective plane over a finite field to construct an almost-disjoint collection of sets with $k=\Theta(n^{1/2})$ and $m=\Theta(n)$. That gives a lower bound. I'm hoping to get an upper bound on $m$ that is better than $m = O(n)$, so this implies I need to restrict attention to the case $c>1/2$.

nonuser
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D.W.
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  • An attempt to address your comments:

    Not sure if this will help, but: Let $\chi_{S_{i}}$ be the characteristic function of $S_{i}$. Your conditions say that (1) for each $q \in {1, 2, \ldots, n}$, $$ 0 \leq \sum_{i=1}^{m} : \chi_{S_{i}}(q) \leq m, $$ (2) for each $S_{i}$, $$ \sum_{1 \leq q \leq n} : \chi_{S_{i}}(q) = k, $$ (3) For each pair $i_{1}, i_{2}$ and each $q$, $$ 0 \leq \chi_{S_{i_{1}}}(q) : \chi_{S_{i_{2}}}(q) \leq 1. $$

     – avs May 11 '17 at 17:09
  • This does not look at all easy. – DanielWainfleet Jul 01 '17 at 23:29

2 Answers2

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Let $P$ be a set of all pairs of elements in $\{1,2,...,n\}$.

Make a bipartite graph between set $P$ and collection of sets $S_1,S_2,...S_m$. Connect set $\{x,y\}$ with set $S_i$ iff $x \in S_i$ and $y\in S_i$. So each pair is connected with at most one set from the collection and each set from the collection is connected with exactly ${k\choose 2}$ pairs. So we have $$m\cdot {k\choose 2} \leq {n\choose 2}\implies \boxed{m\leq {n^2-n\over k^2-k}}$$

D.W.
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nonuser
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  • Awesome! That's just the sort of bound I was hoping for. Thank you. Is this bound tight? Or close to tight? – D.W. Feb 20 '19 at 19:53
  • I don't know, I just find out that equality is achieced iff each element from n-set is in exactly ${n-1\over k-1}$ sets. So equality is possible only if $k-1\mid n-1$. – nonuser Feb 20 '19 at 19:55
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$\quad$No there is not a really an upper bound in some cases. Take k=1, then to share at most 1 in common is trivial, there would be infinitely many 1 element sets, for any n value. With k=2, you can think of connecting distinct pairs of vertices of a regular n-gon. How many pairs are there ? $T_{n-1}$ where T stands for the set of triangular numbers.

$\quad$I suggest looking at the fact, that there are at least k-1 elements, the sets can't have in common. How many disjoint sets of size k-1 are there ? There can be at most, one set, with this as it's elements it doesn't share in our collection. So the upper bound is technically there, but hits infinity when $0<k\le\text{bound}$.

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    The Q is not asking for a fixed upper bound for m as a function of n for a fixed k but an asymptotic functional upper bound. – DanielWainfleet Jul 01 '17 at 23:28
  • okay so what's the asymptotic growth rate of the number of k-1 disjoint subsets as n grows, for constant k ? it can't grow faster than that except on local scales. –  Jul 01 '17 at 23:34
  • I feel like there must be some misunderstanding. There certainly is an upper-bound; for instance, we must have $m \le n^k$. I'm asking for an asymptotic upper bound (hopefully one as close to tight as possible). In particular, I'd like to know the dependence on $n$. Think of this as follows: fix $c$ such that $1/2<c<1$, and then define $g(n)$ to be the largest $m$ such that there exist almost-disjoint sets $S_1,\dots,S_m$ such that $|S_i| \le n^c$ for all $i$. I want to know the asymptotic growth of $g(n)$, as a function of $n$. – D.W. Jul 02 '17 at 02:43
  • O($\lfloor\frac{n}{k-1}\rfloor$) ? –  Jul 02 '17 at 11:56