Counting intersecting and pairs of cross-intersecting families
aa r X i v : . [ m a t h . C O ] O c t Counting intersecting and pairs of cross-intersectingfamilies
Peter Frankl, Andrey Kupavskii ∗ Abstract
A family of subsets of { , . . . , n } is called intersecting if any two of its sets intersect.A classical result in extremal combinatorics due to Erd˝os, Ko, and Rado determinesthe maximum size of an intersecting family of k -subsets of { , . . . , n } . In this paper westudy the following problem: how many intersecting families of k -subsets of { , . . . , n } are there? Improving a result of Balogh, Das, Delcourt, Liu, and Sharifzadeh, wedetermine this quantity asymptotically for n ≥ k + 2 + 2 √ k log k and k → ∞ . More-over, under the same assumptions we also determine asymptotically the number of non-trivial intersecting families, that is, intersecting families for which the intersectionof all sets is empty. We obtain analogous results for pairs of cross-intersecting families. MSc classification: 05D05 A family is a collection of subsets of an n -element set [ n ]. Collections F ⊂ (cid:0) [ n ] k (cid:1) are called k -uniform families . A family F is called intersecting if F ∩ F ′ = ∅ holds for all F, F ′ ∈ F .Similarly, F ⊂ (cid:0) [ n ] k (cid:1) and G ⊂ (cid:0) [ n ] l (cid:1) are called cross-intersecting , if for all F ∈ F , G ∈ G onehas F ∩ G = ∅ . The research concerning intersecting families was initiated by Erd˝os, Ko and Rado, whodetermined the maximum size of intersecting families.
Theorem 1 (Erd˝os, Ko, Rado [3]) . Suppose that n ≥ k > and F ⊂ (cid:0) [ n ] k (cid:1) is intersecting.Then |F | ≤ (cid:18) n − k − (cid:19) . (1)The family of all k -sets containing a fixed element shows that (1) is best possible. Hiltonand Milner proved in a stronger form that for n > k these are the only families on whichthe equality is attained. We say that an intersecting family F is non-trivial if T F ∈F F = ∅ , that is, if it cannot be pierced by a single point. ∗ Moscow Institute of Physics and Technology, Ecole Polytechnique F´ed´erale de Lausanne; Email: [email protected]
Research of Andrey Kupavskii is supported by the grant RNF 16-11-10014. heorem 2 (Hilton, Milner [7]) . Let n > k > and suppose that F ⊂ (cid:0) [ n ] k (cid:1) is a non-trivialintersecting family. Then |F | ≤ (cid:18) n − k − (cid:19) − (cid:18) n − k − k − (cid:19) + 1 . (2)For n = 2 k + 1 the difference between the upper bounds (1) and (2) is only k − n − k increases, this difference gets much larger. The number of subfamiliesof F is 2 |F| , and thus the ratio between the number of subfamilies of the Erd˝os–Ko–Radofamily and that of the Hilton–Milner family is 2 k − for n = 2 k + 1 and grows very fast as n − k increases. This serves as an indication that most intersecting families are trivial , i.e.,satisfy T F ∈F F = ∅ . In an important recent paper Balogh, Das, Delcourt, Liu, and Sharifzadeh [1] provedthis in the following quantitative form. Let I ( n, k ) denote the total number of intersectingfamilies F ⊂ (cid:0) [ n ] k (cid:1) . Theorem 3 (Balogh, Das, Delcourt, Liu, and Sharifzadeh [1]) . If n ≥ k + 8 log k then I ( n, k ) = ( n + o (1))2( n − k − ) , (3) where o (1) → 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 (see Lemma 10). They obtain this bound using thefollowing fundamental result of Bollob´as.
Theorem 4 (Bollob´as [2]) . Suppose that
A ⊂ (cid:0) [ n ] a (cid:1) , B ⊂ (cid:0) [ n ] b (cid:1) with A = { A , . . . , A m } , B = { B , . . . , B m } satisfy A i ∩ B j = ∅ iff i = j . Then m ≤ (cid:18) a + ba (cid:19) . (4)Note that the bound (4) is independent of n . In [2] it is proved in a more general setting,not requiring uniformity, i.e., for A , B ⊂ [ n ] . The uniform version (4) was rediscoveredseveral 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 ofcross-intersecting families. Let us denote by CI ( n, a, b, t ) ( CI ( n, a, b, [ t , t ])) the number ofpairs of cross-intersecting families A ⊂ (cid:0) [ n ] a (cid:1) , B ⊂ (cid:0) [ n ] b (cid:1) with |A| = t ( t ≤ |A| ≤ t ). We alsodenote CI ( n, a, b ) := P t CI ( n, a, b, t ).We prove the following bound for the number of pairs of cross-intersecting families. Theorem 5.
Choose a, b, n ∈ N and put c := max { a, b } , T := (cid:0) n − a + b − n − a (cid:1) . For n ≥ a + b +2 √ c log c + 2 max { , a − b } , a, b → ∞ , and b ≫ log a we have CI ( n, a, b ) =(1 + δ ab + o (1))2( nc ) , (5) CI ( n, a, b, [1 , T ]) =(1 + o (1)) (cid:18) na (cid:19) nb ) − ( n − ab ) , (6) where δ ab = 1 if a = b , and otherwise.
2n fact, (6) easily implies (5) (see Section 2 for details).For a family
F ⊂ (cid:0) [ n ] k (cid:1) we define the diversity γ ( F ) of F to be |F | − ∆( F ), where∆( F ) := max i ∈ [ n ] (cid:12)(cid:12) { F : i ∈ F ∈ F } (cid:12)(cid:12) . For an integer t denote by I ( n, k, t ) ( I ( n, k, ≥ t )) thenumber of intersecting families with diversity t (at least t ). In particular, I ( n, k, ≥
1) is thenumber of non-trivial intersecting families. With the help of (6) we obtain a refinement ofTheorem 3.
Theorem 6.
For n ≥ k + 2 + 2 √ k log k and k → ∞ we have I ( n, k ) =( n + o (1))2( n − k − ) , (7) I ( n, k, ≥
1) =(1 + o (1)) n (cid:18) n − k (cid:19) n − k − ) − ( n − k − k − ) . (8)Again, it is easy to see that (8) implies (7) (see Section 3 for details). In the next sectionwe present the proof of Theorem 5, and in Section 3 we give the proof of Theorem 6. Let us define the lexicographic order on the k -subsets of [ n ]. We have F ≺ G in the lexico-graphic order if min F \ G < min G \ F holds. E.g., { , } ≺ { , } . For 0 ≤ m ≤ (cid:0) nk (cid:1) let L ( k ) ( m ) denote the family of first m k -sets in the lexicographic order. E.g., L ( k ) (cid:0)(cid:0) n − k − (cid:1)(cid:1) = { F ∈ (cid:0) [ n ] k (cid:1) : 1 ∈ F } . Next we state the Kruskal–Katona Theorem [11], [9], which is one of the most importantresults in extremal set theory.
Theorem 7 (Kruskal [11], Katona [9]) . If A ⊂ (cid:0) [ n ] a (cid:1) and B ⊂ (cid:0) [ n ] b (cid:1) are cross-intersectingthen L ( a ) ( |A| ) and L ( b ) ( |B| ) are cross-intersecting as well. Computationwise, the bounds arising from the Kruskal–Katona Theorem are not easy tohandle. Lov´asz [13] found a slightly weaker but very handy form, which may be stated asfollows.
Theorem 8 (Lov´asz [13]) . Let n ≥ a + b , and consider a pair of cross-intersecting families A ⊂ (cid:0) [ n ] a (cid:1) , B ⊂ (cid:0) [ n ] b (cid:1) . If |A| = (cid:0) xn − a (cid:1) for a real number x ≥ n − a , then |B| ≤ (cid:18) nb (cid:19) − (cid:18) xb (cid:19) holds. (9)Note that for x ≥ k − (cid:0) xk (cid:1) is a monotone increasing function of x . Thus x is uniquely determined by |A| and a .We would also need the following result, which proof is based on Theorem 7 and whichis a combination of a result of Frankl and Tokushige [5] (Theorem 2 in [5]) and two resultsof Kupavskii and Zaharov [12] (Part 1 of Theorem 1 and Corollary 1).3 heorem 9 (Frankl, Tokushige, [5], Kupavskii, Zakharov [12]) . Let n > a + b and supposethat the families A ⊂ (cid:0) [ n ] a (cid:1) , B ⊂ (cid:0) [ n ] b (cid:1) are cross-intersecting. If for some real number α ≥ we have (cid:0) n − αn − a (cid:1) ≤ |A| ≤ (cid:0) n − a + b − n − a (cid:1) , then |A| + |B| ≤ (cid:18) nb (cid:19) + (cid:18) n − αa − α (cid:19) − (cid:18) n − αb (cid:19) . (10)Note that the upper bound on |A| in this theorem is exactly the same as in (6).We go on to the proof of Theorem 5. First we show that (6) implies (5). We mayw.l.o.g. assume for this paragraph that c = b ≥ a . For b > a we have T ≥ (cid:0) na (cid:1) and thus CI ( n, a, b, t ) = 0 for t > T . Therefore, we have CI ( n, a, b ) = CI ( n, a, b, CI ( n, a, b, [1 , T ]).If a = b , then T = (cid:0) n − a − (cid:1) , and it follows from Theorem 7 that if A , B ⊂ (cid:0) [ n ] a (cid:1) are cross-intersecting, then min {|A| , |B|} ≤ (cid:0) n − a − (cid:1) . Therefore, in the case a = b we have2 CI ( n, a, b, − ≤ CI ( n, a, b ) ≤ CI ( n, a, b,
0) + CI ( n, a, b, [1 , T ])) . The “-1” in the first inequality stands for a pair of empty families, which is counted twice.At the same time, we have CI ( n, a, b,
0) = 2( nb ). Thus, in both cases b > a and b = a it issufficient to show that the right-hand side of (6) is o (2( nb )). We first note that n − a − b ≥ √ n for b ≥ a , n ≥ a + b + 2 √ b log b , since 4 b log b ≥ a + b + 2 √ b log b . The rest is done by asimple calculation: CI ( n, a, b, [1 , T ])2( nb ) = (cid:18) na (cid:19) − ( n − ab ) ≤ n − ( b + √ nb ) = o (1) . Next, discuss the proof of the lower bound in (6). To obtain that many pairs of inter-secting families, take A := { A } , A ∈ (cid:0) [ n ] a (cid:1) and B ( A ) := { B ∈ (cid:0) [ n ] b (cid:1) : B ∩ A = ∅} . Next,choose an arbitrary subfamily B ⊂ B ( A ). We only need to assure that few of these pairs ofsubfamilies are counted twice. Actually, we count a pair of families twice only in the casewhen a = b and both A , B consist of one set. The number of such pairs is (cid:0) [ n ] a (cid:1) and isnegligible compared to the right hand side of (6).We pass to the proof of the upper bound.2 ≤ |A| ≤ n − a ≤ |A| ≤ n − a ≤ |A| ≤ n − a . Applying Theorem 7, the size of the (unique) maximal family B ′ thatforms a cross-intersecting pair with A is maximized if A consists of two sets A , A thatintersect in a − |B ′ | ≤ (cid:0) nb (cid:1) − (cid:0) n − a +1 b (cid:1) + (cid:0) n − a − b − (cid:1) . Any other family B that forms a cross-intersecting pair with A must be a subfamily of B ′ .So we can bound the number of pairs of cross-intersecting families A , B with 2 ≤ |A| ≤ n − a as follows: P n − at =2 CI ( n, a, b, t )2( nb ) − ( n − ab ) ≤ n − a X t =2 (cid:18)(cid:0) na (cid:1) t (cid:19) nb ) − ( n − a +1 b ) + ( n − a − b − )2( nb ) − ( n − ab ) ≤ n − ( n − a − b − ) = o (1) . − a + 1 ≤ |A| ≤ (cid:0) n − un − a (cid:1) n − a + 1 ≤ |A| ≤ (cid:0) n − un − a (cid:1) n − a + 1 ≤ |A| ≤ (cid:0) n − un − a (cid:1) , where u = √ c log c + max { , a − b } . Note that n − a + 1 = (cid:0) n − a +1 n − a (cid:1) . In this case the bound is similar, but we use Theorem 8 to bound the size of |B| .For A with |A| = t := (cid:0) n − u ′ n − a (cid:1) , where u ≤ u ′ ≤ a −
1, we get |B| ≤ nb ) − ( n − u ′ b ), and since (cid:0) ( na ) t (cid:1) ≤ n t , we have the following bound: CI ( n, a, b, t )2( nb ) − ( n − ab ) ≤ (cid:18)(cid:0) na (cid:1) t (cid:19) nb ) − ( n − u ′ b )2( nb ) − ( n − ab ) ≤ n ( n − u ′ n − a )2 − ( n − u ′− b − ) . At the same time we have n ≥ a + b + 2 u and (cid:0) n − u ′ n − a (cid:1)(cid:0) n − u ′ − b − (cid:1) = n − u ′ b n − a − b − Y i =0 n − b − u ′ − in − a − i ≤ nb n − a − b − Y i =0 n − a − √ c log c − in − a − i ≤ nb e −√ c log c ( P n − ai = b +1 1 i ) ≤ n (11)for sufficiently large c . Indeed, √ c log c P n − ai = b +1 1 i ≥ √ c log c P b +2 √ c log ci = b +1 1 i ≥ (1+ o (1)) √ c log c ) c =(2 + o (1)) log c , which justifies (11) for n ≤ b / . For n > b / we have √ c log c P n − ai = b +1 1 i ≥ (1 + o (1)) √ c log c log nb ≥ (1 + o (1)) √ c log c log n / ≫ log n , which justifies (11) for n > b / .We conclude that ( n − un − a ) X t = n − a +1 CI ( n, a, b, t )2( nb ) − ( n − bb ) ≤ − ( n − u ′− b − ) = o (1) . (cid:0) n − un − a (cid:1) < |A| ≤ T (cid:0) n − un − a (cid:1) < |A| ≤ T (cid:0) n − un − a (cid:1) < |A| ≤ T , where u = √ b log b + max { , a − b } . Using the Bollobas set-pair in-equality, it is not difficult to obtain the following bound on the number of maximal pairs ofcross-intersecting families. Lemma 10.
The number of maximal cross-intersecting pairs A ′ ⊂ (cid:0) [ n ] a (cid:1) , B ′ ⊂ (cid:0) [ n ] b (cid:1) is at most [ (cid:0) na (cid:1)(cid:0) nb (cid:1) ]( a + ba ) . 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 B ′ -generating family M ⊂ A ′ such that B ′ = { B ∈ (cid:0) [ n ] b (cid:1) : B ∩ M = ∅ for all M ∈ M} . We claim that |M| ≤ (cid:0) a + ba (cid:1) . Indeed, due to minimality, for each set M ′ ∈ M the family B ′′ := { B ∈ (cid:0) [ n ] b (cid:1) : B ∩ M = ∅ for all M ∈ M − { M ′ }} strictlycontains B ′ . Therefore, there is a set B in B ′′ \ B ′ such that B ∩ M ′ = ∅ , B ∩ M = ∅ for all M ∈ M − { M ′ } . Applying the inequality (4) to M and the collection of such sets B , weget that |M| ≤ (cid:0) a + bb (cid:1) . 5nterchanging the roles of A ′ and B ′ , we get that a minimal A ′ -generating family hassize at most (cid:0) a + bb (cid:1) as well. Now the bound stated in the lemma is just a crude upper boundon the number of ways one can choose these two generating families out of (cid:0) [ n ] a (cid:1) and (cid:0) [ n ] b (cid:1) ,respectively.Combined with the bound (10) on the size of any maximal pair of families with suchcardinalities, we get that CI ( n, a, b, [ (cid:0) n − un − a (cid:1) , T ])2( nb ) − ( n − ab ) ≤ (cid:20)(cid:18) na (cid:19)(cid:18) nb (cid:19)(cid:21) ( a + ba ) 2( nb ) + ( n − un − a ) − ( n − ub )2( nb ) − ( n − ab ) ≤ n ( a + bb )2( n − un − a ) − ( n − u − b − ) . (12)We also have (cid:0) a + bb (cid:1)(cid:0) n − u − b − (cid:1) = n − ub b − Y i =0 a + b − in − u − i ≤ n (cid:16) a + bn − u (cid:17) b ≤ n . Indeed, the last inequality is clearly valid for n ≥ ( a + b ) , b → ∞ . If n < ( a + b ) , then thebefore-last expression is at most e log n − b ( n − a − b − u ) n − u ≤ e a + b ) − b ( u +max { ,a − b } ) O ( a + b ) ≤ e a + b ) − Ω(min { b,u } ) . Since by the assumption we have b ≫ log( a + b ) and also, obviously, u ≫ log( a + b ), the lastexpression is at most e − a + b ) < n . Taking into account (11), which is valid for u ′ = u , we conclude that the right-hand sideof (12) is o (1). We need a theorem due to Frankl [4], proved in the following, slightly stronger, form in [12].
Theorem 11 ([4, 12]) . Let
F ⊂ (cid:0) [ n ] k (cid:1) be an intersecting family, and n > k . Then, if γ ( F ) ≥ (cid:0) n − u − k − u (cid:1) for some real ≤ u ≤ k , then |F | ≤ (cid:18) n − k − (cid:19) + (cid:18) n − u − k − u (cid:19) − (cid:18) n − u − k − (cid:19) . (13)We go on to the proof Theorem 6. Let us first show that (8) implies (7). Indeed, usingthat n − k − ≥ √ n and k → ∞ in the assumptions of Theorem 6, we get I ( n, k, ≥ n − k − ) ∼ n (cid:18) n − k (cid:19) − ( n − k − k − ) ≤ n +log n − ( k + √ nk − ) = o (1) . I ( n, k, ≥
1) = o (2( n − k − )). On the other hand, it is easy to see that I ( n, k,
0) =( n + o (1))2( n − k − ) (for the proof see [1]).Let us prove the lower bound in (8). For S ∈ (cid:0) [ n ] k (cid:1) , i ∈ [ n ] \ S define the family H ( i, S ) := { S } ∪ { H ∈ (cid:0) [ n ] k (cid:1) : i ∈ H, H ∩ S = ∅} . Due to Theorem 2, these families are the largestnon-trivial intersecting families. We have |H ( i, s ) | = (cid:0) n − k − (cid:1) − (cid:0) n − k − k − (cid:1) + 1 , and each suchfamily contains no less than2( n − k − ) − ( n − k − k − ) − k n − k − ) = (1 + o (1))2( n − k − ) − ( n − k − k − ) (14)non-trivial intersecting subfamilies, as k → ∞ . Indeed, a subfamily of H ( i, S ) containing S is non-trivial unless all sets containing i contain also a fixed j ∈ S . In other words, they mustbe a subset of a family I ( i, j, S ) := { S } ∪ { I ∈ (cid:0) [ n ] k (cid:1) : i, j ∈ S } . The number of subfamiliesof I ( i, j, S ) containing S is 2( n − k − ). Next, we have (cid:0) n − k − (cid:1) − (cid:0) n − k − k − (cid:1) − (cid:0) n − k − (cid:1) ≥ (cid:0) n − k − (cid:1) , and thusthe last inequality in the displayed formula above holds since 2( n − k − ) ≫ k . Denote the set ofall non-trivial subfamilies of H ( i, S ) by ˜ H ( i, S ).Therefore, P S ∈ ( nk ) ,i/ ∈ S | ˜ H ( i, S ) | = (1 + o (1)) n (cid:0) n − k (cid:1) n − k − ) − ( n − k − k − ) . On the other hand, thepairwise intersections of these families are small: the families from ˜ H ( i, S ) ∩ ˜ H ( i, S ′ ) formthe set I ( n, k, I ( n, k,
2) = o ( I ( n, k, i ∈ [ n ] and F ⊂ (cid:0) [ n ] k (cid:1) we use the standard notation F ( i ) := { F − { i } : i ∈ F ∈ F } ⊂ (cid:18) [ n ] − { i } k − (cid:19) , F (¯ i ) := { F ∈ F : i / ∈ F } ⊂ (cid:18) [ n ] − { i } k (cid:19) . Note that if F is intersecting then F ( i ) and F (¯ i ) are cross-intersecting.We count the number of families with different diversity separately. The number offamilies F with i being the most popular element and γ ( F ) ≤ (cid:0) n − k − (cid:1) is at most the numberof cross-intersecting pairs F (¯ i ) , F ( i ).Therefore, we may apply (6) with n ′ := n − , a := k, b := k −
1, and get that the numberof such families F is at most (1+ o (1)) (cid:0) n − k (cid:1) n − k − ) − ( n − k − k − ). Note that n ′ ≥ a + b +2 √ a log a +2and, in terms of Theorem 5, we have T = (cid:0) n − k − (cid:1) for our case. Multiplying the number ofsuch families by the number of choices of i , we get the claimed asymptotic.We are only left to prove that there are few families with diversity larger than (cid:0) n − k − (cid:1) .Using the upper bound (cid:0) nk (cid:1) ( k − k − ) for the number of maximal intersecting families in (cid:0) [ n ] k (cid:1) obtained in [1] (see Lemma 10 for the proof of a similar statement), combined with the7ound (13) on the size of any maximal family with such diversity, we get that I ( n, k, ≥ (cid:0) n − k − (cid:1) )2( n − k − ) − ( n − k − k − ) ≤ (cid:18) nk (cid:19) ( k − k − ) 2( n − k − ) + ( n − k − ) − ( n − k − )2( n − k − ) − ( n − k − k − ) ≤ n ( k − k − )2( n − k − ) − ( n − k − ) . (15)Putting n = 2 k + x , we have (cid:0) n − k − (cid:1) / (cid:0) n − k − (cid:1) = ( n − k − n − k − n − k − ≤ (2 k + x ) k ( k + x − ≤ − x ( k + x ) ≤ − k . On the other hand, (cid:0) k − k − (cid:1)(cid:0) n − k − (cid:1) = n − k − k Y i =1 k − in − − i ≤ n (cid:16) kn − (cid:17) k ≤ kn , where the last inequality is clearly valid for n ≥ k + 2 + 2 √ k log k and sufficiently large k .We conclude that the right-hand side of (15) is at most 2 k ( n − k − ) = o (1). References [1] J. Balogh, SA. Das, M. Delcourt, H. Liu, M. Sharifzadeh,
Intersecting families of dicretestructures 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 QuarterlyJournal of Mathematics, 12 (1961) N1, 313–320.[4] P. Frankl,
Erdos–Ko–Rado theorem with conditions on the maximal degree , Journal ofCombinatorial Theory, Series A 46 (1987), N2, 252–263.[5] P. Frankl, N. Tokushige,
Some best possible inequalities concerning cross-intersectingfamilies , 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), unpub-lished 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 , CRAcad. 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 ofCombinatorial Theory, Series A 17 (1974), N2, 265–266.811] 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. Lov´asz,