Counting intersecting and pairs of cross-intersecting families

0.13 MB·

by  Peter Frankl

Counting intersecting and pairs of cross-intersecting
families
7 Peter Frankl, Andrey Kupavskii
∗
1
0
2

n
a Abstract
J

5 A family of subsets of 1,...,n is called intersecting if any two of its sets intersect.
{ }
1 A classical result in extremal combinatorics due to Erd˝os, Ko, and Rado determines

the maximum size of an intersecting family of k-subsets of 1,...,n . In this paper
]
O we study the following problem: how many intersecting fam{ilies of k-}subsets of n are
C there? Improving the result of Balogh et al., we determine the asymptotic of this
. quantity for n 2k+2+2√klogk and k . Moreover, in the same assumptions
h
≥ → ∞
t we also determine the asymptotic of the number of non-trivial intersecting families,
a
m that the intersecting families in which the intersection of all sets is empty. We obtain
[ analogous results for pairs of cross-intersecting families.

1
v
1 Introduction
0
1
1
Let [n] = 1,...,n be the standard n-element set and for an integer 0 k n let [n]
4
{ } ≤ ≤ k
0 denote all its k-element subsets. Subsets [n] are called k-uniform families. (cid:0) (cid:1)
. F ⊂ k
1 A family is called intersecting if F F(cid:0) =(cid:1) holds for all F,F . Similarly, if
′ ′
0 F ∩ 6 ∅ ∈ F
[n] and [n] then they are called cross-intersecting if for all F , G one
7
F ⊂ k G ⊂ l ∈ F ∈ G
1 has F(cid:0) (cid:1)G = . (cid:0) (cid:1)
: ∩ 6 ∅
v For i [n] and [n] we use the standard notation
i ∈ F ⊂ k
X (cid:0) (cid:1)
[n] i
r
(i) := F i : i F −{ }
a
F { −{ } ∈ ∈ F} ⊂ (cid:18) k 1 (cid:19)
−
[n] i
(¯i) := F : i / F −{ } .
F { ∈ F ∈ } ⊂ (cid:18) k (cid:19)
Note that if is intersecting then (i) and (¯i) are cross-intersecting.
F F F
The research concerning intersecting families was initiated by Erd˝os, Ko and Rado who
determined the maximum size of intersecting families.
∗Moscow Institute of Physics and Technology, Ecole Polytechnique F´ed´erale de Lausanne; Email:
[email protected] Research supported in part by the ANR project STINT under reference ANR-
13-BS02-0007and by the grant N 15-01-03530of the Russian Foundation for Basic Research.
1
Theorem A (Erdo˝s, Ko, Rado [3]). Suppose that n 2k > 0 and [n] is intersecting.
≥ F ⊂ k
Then (cid:0) (cid:1)
n 1
− . (1)
|F| ≤ (cid:18)k 1(cid:19)
−
the family of all k-sets containing a fixed element shows that (1) is best possible. Hilton
and Milner proved in a stronger form that for n > 2k these are the only families on which
the equality is attained.
Theorem B (Hilton, Milner [7]). Let n > 2k > 0 and suppose that [n] is intersecting
F ⊂ k
but F = . Then (cid:0) (cid:1)
F ∅
T ∈F n 1 n k 1
− − − +1. (2)
|F| ≤ (cid:18)k 1(cid:19)−(cid:18) k 1 (cid:19)
− −
For n = 2k+1 the difference between the upper bounds is only k 1. However, as n 2k
− −
increases, this difference gets much larger. Considering the number of subfamilies 2 of
|F|
, the above difference turns into ratio and serves as an indication that most intersecting
F
families are trivial, i.e., satisfy F = .
F 6 ∅
In an important recent papTer B∈Falogh et al. [1] proved this in the following quantitative
form. Let I(n,k) denote the total number of intersecting families [n] .
F ⊂ k
(cid:0) (cid:1)
Theorem C (Balogh et al [1]). If n 3k +8logk then
≥
(n 1)
I(n,k) = (n+o(1))2 k−1 , (3)
−
where o(1) 0 as k .
→ → ∞
One of the main tools of the proof of (3) is a nice bound on the number of maximal (i.e.,
non-extendable) intersecting families. We are going to explain it later. Let us simply point
out that they obtain these bounds using the following fundamental result of Bolloba´s.
Theorem D (Bollob´as [2]). Suppose that [n] , [n] with = A ,...,A , =
A ⊂ a B ⊂ b A { 1 m} B
B ,...,B satisfy A B = iff i = j. Then(cid:0) (cid:1) (cid:0) (cid:1)
1 m i j
{ } ∩ ∅
a+b
m . (4)
≤ (cid:18) a (cid:19)
Note that the bound (4) is independent of n. In [2] it is proved in a more general setting,
not requiring uniformity, i.e., , 2[n]. The uniform version (4) was rediscovered several
A B ⊂
years later by Jaeger-Payan [8] and Katona [10].
We are going to use (4) to obtain an upper bound on the number of maximal pairs of
cross-intersecting families. Let us denote by CI(n,a,b,t) (CI(n,a,b,[t ,t ])) the number of
1 2
pairs of cross-intersecting families [n] , [n] and = t (t t ). We also
A ⊂ a B ⊂ b |A| 1 ≤ |A| ≤ 2
denote CI(n,a,b) := CI(n,a,b,t). (cid:0) (cid:1) (cid:0) (cid:1)
t
We prove the folloPwing bound for the number of pairs of cross-intersecting families:
2
Theorem 1. Choose a,b,n N and put c := max a,b , T := n a+b 1 . For n a+b+
− −
∈ { } n a ≥
2√clogc+2max 0,a b , a,b , and b loga we have (cid:0) − (cid:1)
{ − } → ∞ ≫
(n)
CI(n,a,b) =(1+δab +o(1))2 c , (5)
n
(n) (n a)
CI(n,a,b,[1,T]) =(1+o(1)) 2 b − −b , (6)
(cid:18)a(cid:19)
where δ = 1 if a = b and 0 otherwise.
ab
Note that if a = b, then twice the right hand side of (6) gives the number of pairs of
cross-intersecting families with both families being nonempty, and if a < b, then the RHS of
(6) gives the number of cross-intersecting pairs with being nonempty.
A
Let us show that (6) implies (5). Assuming for this paragraph that c = b a, one notes
≥
(n)
that CI(n,a,b) = δab(CI(n,a,b,0)+CI(n,a,b,[1,T])) and that CI(n,a,b,0) = 2 b . Thus,
(n)
it is sufficient to show that the right hand side of (6) is o(2 b ). It is done by a simple
calculation:
n
2−(n−ba) 2n−(b+b√n) = o(1).
(cid:18)a(cid:19) ≤
For a family [n] we define the diversity γ( ) of to be ∆( ), where
F ⊂ k F F |F| − F
∆( ) := max F :(cid:0)i (cid:1)F . For an integer t denote by I(n,k,t) (I(n,k, t)) the
i [n]
F ∈ { ∈ ∈ F} ≥
number of intersec(cid:12)ting families with(cid:12)diversity t (at least t). In particular, I(n,k, 1) is the
(cid:12) (cid:12) ≥
number of so-called non-trivial intersecting families, that is, families that cannot be pierced
by a single point. With the help of (6) we obtain a refinement of Theorem C.
Theorem 2. For n 2k +2+2√klogk and k we have
≥ → ∞
(n 1)
I(n,k) =(n+o(1))2 k−1 , (7)
−
n 1
(n 1) (n k 1)
I(n,k, 1) =(1+o(1))n − 2 k−1 − −k −1 . (8)
≥ (cid:18) k (cid:19) − −
Again, it is easy to see that (8) implies (7). Indeed, in the assumptions of the theorem
n 1 k 1
n(cid:18) −k (cid:19)2−(n−k−k−11) ≤ 2n+1−(cid:0)n−kk(cid:1) − ≤ 2n+1−2n−2k = o(1).
(n 1)
Therefore, I(n,k, 1) = o(2 k−1 ). On the other hand, it is easy to see that I(n,k,0) =
≥ −
(n 1)
(n+o(1))2 k−1 (for the easy proof cf. [1]).
−
In the next section we present the proof of (6), and in Section 3 we give the proof of (8).
3
2 Cross-intersecting families
Let us define the lexicographic order on the k-subsets of [n]. We have F G in the lexico-
≺
graphic order if minF G < minG F holds. E.g., 1,10 2,3 . For 0 m n let
\ \ { } ≺ { } ≤ ≤ k
(k)(m) denote the family of first m k-sets in the lexicographic order. E.g., (k) n(cid:0)1(cid:1) =
−
L L k 1
F [n] : 1 F . (cid:0)(cid:0) − (cid:1)(cid:1)
{ ∈ k ∈ }
Ne(cid:0)xt (cid:1)we state the Kruskal-Katona Theorem [11], [9], which is one of the most important
results in extremal set theory.
Theorem E (Kruskal, [11], Katona, [9]). If [n] and [n] are cross-intersecting
A ⊂ a B ⊂ b
then (a)( ) and (b)( ) are cross-intersecting a(cid:0)s w(cid:1)ell. (cid:0) (cid:1)
L |A| L |B|
Computationwise, the bounds arising from the Kruskal-Katona Theorem are not easy to
handle. Lova´sz [13] found a slightly weaker but very handy form, which may be stated as
follows:
Theorem F (Lova´sz, [13]). If n a+b and [n] and [n] are cross-intersecting,
≥ A ⊂ a B ⊂ b
and = x for a real number x n a, then(cid:0) (cid:1) (cid:0) (cid:1)
|A| n a ≥ −
(cid:0) − (cid:1)
n x
holds. (9)
|B| ≤ (cid:18)b(cid:19)−(cid:18)b(cid:19)
Note that for x k 1 the polynomial x is a monotone increasing function of x. Thus
≥ − k
x is uniquely determined by and a. (cid:0) (cid:1)
|A|
We would also need the following result, whose proof is based on Theorem E and which
is a combination of a result of Frankl and Tokushige [5] (Theorem 2 in [5]) and two results
of Kupavskii and Zaharov [12] (Part 1 of Theorem 1 and Corollary 1).
Theorem G (Frankl, Tokushige, [5], Kupavskii, Zakharov [12]). Let n > a+b and suppose
that the families [n] , [n] are cross-intersecting. If for some real number α 1
A ⊂ a B ⊂ b ≥
we have n α (cid:0) n(cid:1)a+b 1 (cid:0), th(cid:1)en
− − −
n a ≤ |A| ≤ n a
(cid:0) − (cid:1) (cid:0) − (cid:1)
n n α n α
+ + − − . (10)
|A| |B| ≤ (cid:18)b(cid:19) (cid:18)a α(cid:19)−(cid:18) b (cid:19)
−
We go on to the proof of Theorem 1. First we discuss the proof of the lower bound
in (5). To obtain that many pairs of intersecting families, take := A ,A [n] and
A { } ∈ a
(A) := B [n] : B A = . Next, choose an arbitrary subfamily (A). (cid:0) (cid:1)
B { ∈ b ∩ 6 ∅} B ⊂ B
The only th(cid:0)ing(cid:1) one has to show is that few of these pairs of subfamilies correspond to
two (A) with different A. Actually, the number of pairs of such subfamilies is equal to
B
I(n,a,b,2), and in the proof of the upper bound we show that I(n,a,b,2) = o(I(n,a,b,1)).
222 nnn aaa. Applying Theorem E, the size of the (unique) maximal family that
′
≤≤≤|||AAA|||≤≤≤ −−− B
forms a cross-intersecting pair with is maximized if consists of two sets A ,A that
1 2
A A
4
intersect in a 1 elements. Therefore, n n a+1 + n a 1 . Any other family
− |B′| ≤ b − −b −b −2 B
that forms a cross-intersecting pair with mu(cid:0)st(cid:1)be (cid:0)a subfa(cid:1)mil(cid:0)y o−f (cid:1).
′
A B
So we can bound the number of pairs of cross-intersecting families , with 2
A B ≤ |A| ≤
n a as follows:
−
tn=−2aCI(n,a,b,t) n−k−1 na 2(nb)−(n−ba+1)+(n−b−a−21) 2n22−(n−ba−11) = o(1).
P (n) (n a) ≤ (cid:18)(cid:0)t(cid:1)(cid:19) (n) (n a) ≤ −
2 b − −b Xt=2 2 b − −b
nnn aaa+++111 nnn−−−uuu , where u = √clogc + max 0,a b . Note that n a + 1 =
−−− ≤≤≤|||AAA|||≤≤≤ nnn aaa { − } −
n a+1 . In this case t(cid:0)(cid:0)(cid:0)he−−−b(cid:1)(cid:1)(cid:1)ound is similar, but we use Theorem F to bound the size of .
−
n a |B|
F(cid:0)or−A(cid:1)with |A| = t := nn−ua′ , where u ≤ u′ ≤ a − 1, we get |B| ≤ 2(nb)−(n−bu′), and since
(na) 2nt, we have the(cid:0)fo−llo(cid:1)wing bound:
t ≤
(cid:0) (cid:1)
CI(n,a,b,t) na 2(nb)−(n−bu′) 2n(nn−ua′)2−(n−bu′1−1).
(n) (n a) ≤ (cid:18)(cid:0)t(cid:1)(cid:19) (n) (n a) ≤ − −
2 b − −b 2 b − −b
At the same time we have n a+b+2u and
≥
(cid:0)n−nnb−u−′u1a−′(cid:1)1 = n−b u′ n−Yia=−0b−1 n−n−b−au−′ −i i ≤
(cid:0) − (cid:1)
n a b 1
n − − − n−a−√clogc−i ne−√clogc(Pni=−ba+1 1i) 1 (11)
b n a i ≤ b ≤ 2n
Yi=0 − −
forsufficientlylargec. Indeed, √clogc n−a 1 √clogc b+2√clogc 1 (1+o(1))2(√clogc)2 =
i=b+1 i ≥ i=b+1 i ≥ c
(2+o(1))logc, which justifies (11) forPn b3/2. For n >Pb3/2 we have √clogc n−a 1
≤ i=b+1 i ≥
(1+o(1))√clogclog n (1+o(1))√clogclogn2/3 logn, which justifies (11) fPor n > b3/2.
b ≥ ≫
We conclude that
(n u)
n−−a CI(n,a,b,t) 2−21(n−bu′1−1) = o(1).
(n) (n b) ≤ −
t=nXa+1 2 b − −b
−
nnn−−−uuu <<< TTT, where u = √blogb + max 0,a b . Using the Bollobas set-pair in-
nnn aaa |||AAA|||≤≤≤ { − }
equ(cid:0)(cid:0)(cid:0)ali−−−ty(cid:1)(cid:1),(cid:1)it is not difficult to obtain the following bound on the number of maximal pairs of
cross-intersecting families.
Lemma 3. The number of maximal cross-intersecting pairs [n] , [n] is at most
A′ ⊂ a B′ ⊂ b
[ n n ](a+ab). (cid:0) (cid:1) (cid:0) (cid:1)
a b
(cid:0) (cid:1)(cid:0) (cid:1)
5
We note that the proof is very similar to the proof of an analogous statement for inter-
secting families from [1].
Proof. Find a minimal -generating family such that = B [n] : B M =
B′ M ⊂ A′ B′ { ∈ b ∩ 6
for all M . We claim that a+b . Indeed, due to minimali(cid:0)ty,(cid:1)for each set
∅ ∈ M} |M| ≤ a
M the family := B [n] : (cid:0)B (cid:1)M = for all M M strictly
′ ∈ M B′′ { ∈ b ∩ 6 ∅ ∈ M − { ′}}
contains . Therefore, there is a set(cid:0)B (cid:1)in such that B M = ,B M = for all
′ ′′ ′ ′
B B \B ∩ ∅ ∩ 6 ∅
M M . Applying the inequality (4) to and the collection of such sets B, we
′
∈ M −{ } M
get that a+b .
|M| ≤ b
Interchanging(cid:0) th(cid:1)e roles of and , we get that a minimal -generating family has
′ ′ ′
A B A
size at most a+b as well. Now the bound stated in the lemma is just a crude upper bound
b
on the numb(cid:0)er o(cid:1)f ways one can choose these two generating families out of [n] and [n] ,
a b
respectively. (cid:0) (cid:1) (cid:0) (cid:1)
Combined with the bound (10) on the size of any maximal pair of families with such
cardinalities, we get that
CI(n,(an,)b,(≥n a(cid:0))nn−−ua(cid:1)) ≤ (cid:18)na(cid:19)(cid:18)nb(cid:19) (a+ab)2(nb)+(n()nn−−ua()n−a()n−bu) ≤ 22n(a+bb)2(nn−−ua)−(n−b−u1−1). (12)
2 b − −b (cid:2) i 2 b − −b
We also have
a+b b 1
n u − a+b i a+b b 1
b = − − n .
n(cid:0) u (cid:1)1 b n u i ≤ n u ≤ 4n
−b −1 Yi=0 − − (cid:16) − (cid:17)
(cid:0) − (cid:1)
Indeed, the last inequality is clearly valid for n (a+b)2,b . If n < (a+b)2, then the
≥ → ∞
before-last expression is at most
elogn−b(n−na−−ub−u) ≤ e2log(a+b)−b(u+mOa(xa{+0b,)a−b}) ≤ e2log(a+b)−Ω(min{b,u}).
Since by the assumption we have b log(a+b) and also, obviously, u log(a+b), the last
≫ ≫
expression is at most e 4log(a+b) < 1 .
− 4n
Taking into account (11), which is valid for u = u, we conclude that the right hand side
′
of (12) is o(1).
3 Intersecting families
We need a theorem due to Frankl [4], proved in the following, slightly stronger, form in [12].
Theorem H ([4, 12]). Let [n] be an intersecting family, and n > 2k. Then, if
F ⊂ k
γ( ) n u 1 for some real 3 u(cid:0) (cid:1)k, then
− −
F ≥ k u ≤ ≤
(cid:0) − (cid:1)
n 1 n u 1 n u 1
− + − − − − . (13)
|F| ≤ (cid:18)k 1(cid:19) (cid:18) k u (cid:19)−(cid:18) k 1 (cid:19)
− − −
6
We go on to the proof Theorem 2. Let us first prove the lower bound. For S [n] ,i
∈ k ∈
[n] S define the family (i,S) := S H [n] : i H,H S = . Due to The(cid:0)ore(cid:1)m B,
\ H { }∪{ ∈ k ∈ ∩ 6 ∅}
these families are the largest non-trivial interse(cid:0)cti(cid:1)ng families. We have (i,s) = n 1
−
|H | k 1 −
n k 1 +1, and each such family contains no less than (cid:0) − (cid:1)
− −
k 1
(cid:0) − (cid:1)
(n 1) (n k 1) (n 2) (n 1) (n k 1)
2 k−1 − −k −1 k2 k−2 = (1+o(1))2 k−1 − −k −1 (14)
− − − − − −
non-trivial intersecting subfamilies, as k . Indeed, a subfamily of (i,S) containing S
→ ∞ H
is non-trivial unless all sets containing i contain also a fixed j S. In other words, they must
∈
be a subset of a family (i,j,S) := S I [n] : i,j S . The number of subfamilies
I { }∪{ ∈ k ∈ }
of (i,j,S) containing S is 2(nk−22). Next, we have(cid:0) n(cid:1)−1 n−k−1 n−2 n−3 , and thus
I − k 1 − k 1 − k 2 ≥ k 2
(cid:0) − (cid:1) (cid:0) − (n(cid:1) 3)(cid:0) − (cid:1) (cid:0) − (cid:1)
the last inequality in the displayed formula above holds since 2 k−2 k. Denote the set of
− ≫
all non-trivial subfamilies of (i,S) by ˜(i,S).
H H
Therefore, S∈(nk),i∈/S|H˜(i,S)| = (1+o(1))n n−k1 2(nk−−11)−(n−k−k−11). On the other hand, the
pairwise intersPections of these families are small(cid:0): th(cid:1)e families from ˜(i,S) ˜(i,S ) form
′
H ∩H
the set I(n,k,2), and we do (somewhat implicitly) show in the proof that I(n,k,2) =
o(I(n,k,1)). It could also be verified by a simple direct, but somewhat tedious calculation.
Therefore, the lower bound is justified.
Next we prove the upper bound. We count the number of families with different diversity
separately. The number of families with i being the most popular element and γ( )
F F ≤
n 3 is at most the number of cross-intersecting pairs (¯i), (i) (recall the definition from
−
k 2 F F
(cid:0)th−e i(cid:1)ntroduction).
Therefore, we may apply (6) with n := n 1,a := k,b := k 1, and get that the number
′
− −
ofsuch familiesF isatmost (1+o(1)) n−k1 2(nk−−11)−(n−k−k−11). Notethat n′ ≥ a+b+2√aloga+2
and, in terms of Theorem 1, we have(cid:0) T (cid:1)= n 3 for our case. Multiplying the number of
−
k 2
such families by the number of choices of i, w(cid:0)e−g(cid:1)et the claimed asymptotic.
We are only left to prove that there are few families with diversity larger than n 3 .
−
k 2
(2k 1) (cid:0) − (cid:1)
Using the upper bound n k−1 for the number of maximal intersecting families in [n]
k − k
obtained in [1] (see Lemm(cid:0)a(cid:1)3 for the proof of a similar statement), combined with the bo(cid:0)und(cid:1)
(13) on the size of any maximal family with such diversity, we get that
I((nn,k1),≥(n(cid:0)nkk−−231(cid:1))) ≤ (cid:18)nk(cid:19)(2kk−−11)2(nk−−(n11)+1)(nk−−(n34)−k(nk1)−−14) ≤ 2n(2kk−−11)2(nk−−34)−(nk−−25). (15)
2 k−1 − −k −1 2 k−1 − −k −1
− − − −
Putting n = 2k +x, we have nk−34 / nk−25 = (n (nk−14))((nk−2k) 2) ≤ ((k2+kx+x2)k)2 ≤ 1− (k+x2x)2 ≤ 1− k1.
On the other hand, (cid:0) − (cid:1) (cid:0) − (cid:1) − − − − −
2k 1 k
− n 4 2k i 2k k 1
(cid:0)nk−15(cid:1) = k −1 n 3− i ≤ n n 3 ≤ 2kn,
k−2 − Yi=1 − − (cid:16) − (cid:17)
(cid:0) − (cid:1)
7
where the last inequality is clearly valid for n 2k +2+2√klogk and sufficiently large k.
≥
1 (n 5)
We conclude that the right hand side of (15) is at most 22k k−2 = o(1).
−
References
[1] J. Balogh, SA. Das, M. Delcourt, H. Liu, M. Sharifzadeh, Intersecting families of dicrete
structures are typically trivial, J. Combinatorial Theory Ser. A 132 (2015), 224–245.
[2] B. Bollob´as, On generalized graphs, Acta Math. Acad. Sci. Hungar. 16 (1965), 447–452.
[3] P. Erd˝os, C. Ko, R. Rado, Intersection theorems for systems of finite sets, The Quarterly
Journal of Mathematics, 12 (1961) N1, 313–320.
[4] P. Frankl, Erdos-Ko-Rado theorem with conditions on the maximal degree, Journal of
Combinatorial Theory, Series A 46 (1987), N2, 252–263.
[5] P. Frankl, N. Tokushige, Some best possible inequalities concerning cross-intersecting
families, Journal of Combinatorial Theory, Series A 61 (1992), N1, 87–97.
[6] A.J.W. Hilton, The Erd˝os-Ko-Rado theorem with valency conditions, (1976), unpublished
manuscript.
[7] A.J.W. Hilton, E.C. Milner, Some intersection theorems for systems of finite sets, Quart.
J. Math. Oxford 18 (1967), 369–384.
[8] F. Jaeger, C. Payan, Nombre maximal d’aretes d’un hypergraphe critique de rang h, CR
Acad. Sci. Paris 273 (1971), 221-223.
[9] G.O.H. Katona, A theorem of finite sets, “Theory of Graphs, Proc. Coll. Tihany, 1966”,
Akad, Kiado, Budapest, 1968; Classic Papers in Combinatorics (1987), 381–401.
[10] G.O.H. Katona, Solution of a problem of A. Ehrenfeucht and J. Mycielski, Journal of
Combinatorial Theory, Series A 17 (1974), N2, 265–266.
[11] J.B. Kruskal, The Number of Simplices in a Complex, Mathematical optimization tech-
niques 251 (1963), 251–278.
[12] A. Kupavskii, D. Zakharov, Regular bipartite graphs and intersecting families,
arXiv:1611.03129
[13] L. Lova´sz, Combinatorial Problems and Exercises, 13.31, North-Holland, Amsterdam
(1979).
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