The number of maximal left-compressed intersecting families

A family of sets $\mathcal A \subseteq [n]^{(r)}$ (subsets of $\{1, \ldots, n\}$ of size $r$ ) is intersecting if every pair of sets in $\mathcal A$ have a common element. If $n < 2r$ then every pair of sets intersect, so $|\mathcal A|$ can be as large as $\binom n r$ . If $n \geq 2r$ then the Erdős–Ko–Rado theorem states that $|\mathcal A| \leq \binom {n-1} {r-1}$ , which (up to relabelling of the ground set) is attained only by the star $\mathcal S$ of all sets containing the element $1$ .

A hands on proof of the Erdős–Ko–Rado theorem use a tool called compression. A family $\mathcal A$ is left-compressed if for every $A \in \mathcal A$ , any set obtained from $A$ by deleting an element and replacing it by a smaller one is also in $\mathcal A$ . You can show by repeatedly applying a certain compression operator that for every intersecting family $\mathcal A$ there is a left-compressed intersecting family $\mathcal A'$ of the same size. Thus it suffices to prove the Erdős–Ko–Rado theorem for left-compressed families, which is easy to do by induction.

There is a strong stability result for large intersecting families. The Hilton–Milner family consists of all sets that contain $1$ and at least one element of $[2,r+1]$ , together with $[2,r+1]$ itself. This is an intersecting family, and in fact is the largest intersecting family not contained in a star. The Hilton–Milner family has size $O(n^{r-2})$ , so any family that gets anything like close to the Erdős–Ko–Rado bound must be a subset of a star.

As part of an alternative proof of the Hilton–Milner theorem, Peter Borg partially answered the following question.

Let $\mathcal A \subseteq [n]^{(r)}$ be an intersecting family and let $X \subseteq [n]$ . Let $\mathcal A(X) = \{A \in \mathcal A : A \cap X \neq \emptyset\}$ . For which $X$ is $|\mathcal A(X)| \leq |\mathcal S(X)|$ ?

Borg used that fact that this is true for $X = [2,r+1]$ to reprove the Hilton–Milner theorem. In Maximum hitting for $n$ sufficiently large I completed the classification of $X$ for which this is true for large $n$ . The proof used the apparently new observation that, for $n \geq 2r$ , every maximal left-compressed intersecting family in $[n]^{(r)}$ corresponds to a unique maximal left-compressed intersecting family of $[2r]^{(r)}$ . In particular, the number of maximal left-compressed intersecting families for $n \geq 2r$ is independent of $n$ . For $r=1, 2, 3, 4, 5, 6$ there are $1, 2, 6, 72, 37145, 1081162102034$ (OEIS) such families respectively. In the rest of this post I’ll explain how I obtained these numbers.

We want to count maximal left-compressed intersecting families of $[2r]^{(r)}$ . The maximal part is easy: the only way to get two disjoint sets of size $r$ from $[2r]$ is to take a set and its complement, so we must simply choose one set from each complementary pair. To make sure the family we generate in this way is left-compressed we must also ensure that whenever we choose a set $A$ we must also choose every set $B$ with $B \leq A$ , where $B \leq A$ means “ $B$ can be obtained from $A$ by a sequence of compressions”. The compression order has the following properties.

If $A = \{a_1 < \cdots < a_r\}$ and $B = \{b_1 < \cdots < b_r\}$ then $A \leq B$ if and only if $a_i \leq b_i$ for each $i$ .
$A \leq B$ if and only if $B^c \leq A^c$ .

Here’s one concrete algorithm.

Generate a list of all sets from $[2r-1]^{(r)}$ . This list has one set from each complementary pair.
Put all $A$ from the list with $A < A^c$ into $\mathcal A$ . (These sets must be in every maximal left-compressed intersecting family.) Note that we never have $A^c < A$ as $A^c$ contains $2r$ but $A$ doesn’t.
Let be the first element of the list and branch into two cases depending on whether we take or .
- If we take $A$ , also take all $B$ from the list with $B < A$ and $B^c$ for all $B$ from the list with $B^c < A$ . (Since $B^c$ contains $2r$ and $A$ doesn’t, the second condition will never actually trigger.)
- If we take $A^c$ , also take all $B$ from the list with $B < A^c$ and $B^c$ for all $B$ from the list with $B^c < A^c$ . (It is cheaper to test $A < B$ than the second condition to avoid taking complements.)
Repeat recursively on each of the two lists generated in the previous step. Stop on each branch whenever the list of remaining options is empty.

The following is a fairly direct translation of this algorithm into Haskell that makes no attempt to store the families generated and just counts the number of possibilities. A source file with the necessary import’s and the choose function is attached to the end of this post.

r = 5

simpleOptions = [a | a <- choose r [1..(2*r-1)], not [dollar-sign] a `simpleLeftOf` (simpleComplement a)]

simpleLeftOf xs ys = all id [dollar-sign] zipWith (<=) xs ys

simpleComplement a = [1..(2*r)] \\ a

simpleCount [] = 1
simpleCount (a:as) = simpleCount take + simpleCount leave
  where
    -- take a
    -- all pairs with b < a or b^c < a are forced
    -- second case never happens as b^c has 2r but a doesn't
    take = [b | b <- as, not [dollar-sign] b `simpleLeftOf` a]
    -- leave a, and so take a^c
    -- all pairs with b < a^c or b^c < a^c (equivalently, a < b) are forced
    c = simpleComplement a
    leave = [b | b <- as, not (b `simpleLeftOf` c || a `simpleLeftOf` b)]

This will compute the number of maximal left-compressed intersecting families for $r \leq 5$ in a fraction of a second. For $r=6$ it would probably find the answer in less than a month. I obtained the value for $r=6$ in a couple of days on a single core by using a better representation of the sets in our family.

The dream is to pack all of the elements of our list into a single machine word and perform each comparison in a small number of instructions. For example, we could encode an element of $[12]^{(6)}$ by writing each element as $4$ binary digits then concatenating them in increasing order to obtain a 24 bit word. But comparing two such words as integers compares the corresponding sets lexicographically rather than pointwise. Edward Crane suggested that as the lists are so short and the elements are so small we can afford to be quite a lot more wasteful in our representation: we can write each element of our set in unary! The rest of this section should be considered joint work with him.

The first iteration of the idea is to write each element $x$ of $[12]$ as a string of $x$ 1’s followed by $12-x$ 0’s, then concatenate these strings to obtain a representation of our set. This representation has the great advantage that we can compare sets pointwise by comparing strings bitwise, and we can do this using very few binary operations: $a$ is contained in $b$ if and only if $a \& b = a$ .

Unfortunately this representation uses 72 bits in total, so won’t fit into a 64-bit machine word. Observing that we never use $0$ and encoding by $x-1$ 1‘s followed by $11-x$ 0‘s saves only 6 bits. But we can do even better by encoding each element of the set differently. The first element is always at least 1, the second is always at least 2 and so on. Similarly, the first element is at most 7, the second at most 8 and so on. Working through the details we arrive at the following representation.

Identify each element of $[12]^{(6)}$ by an “up-and-right” path from the bottom left to the top right corner of a $6 \times 6$ grid: at the $i$ th step move right if $i$ is in your set and up if it isn’t. Then $A \leq B$ if and only if the path corresponding to $A$ never goes below the path corresponding to $B$ . So we can compare sets by comparing the regions below the corresponding paths. Recording these regions can be done using 36 bits, which happily sits inside a machine word. This representation also has the helpful property that taking the complement of a set corresponds to reflecting a path about the up-and-right diagonal, so the representation of the complement of a set can be obtained by swapping certain pairs of bits followed by a bitwise NOT.

The value for $r=6$ was obtained using this new representation and the old algorithm, with one minor tweak. It’s a bad idea to start with a lexicographically ordered list of sets, as the early decisions will not be meaningful and not lead to much of a reduction in the length of the the lists. Optimal selection of which pair to decide at each stage is probably a complicated question. As a compromise I randomised the order of the list at the start of the process, then took the first remaining pair at each stage.

The Haskell source is here. There are a few more performance tricks to do with the exact bit representation of the sets, which I’m happy to discuss if anything is unclear.

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