On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method
aa r X i v : . [ m a t h . C O ] M a r On the number of maximal intersecting k -uniformfamilies and further applications of Tuza’s set pairmethod Zolt´an L´or´ant Nagy
MTA–ELTE Geometric and Algebraic Combinatorics Research GroupH–1117 Budapest, P´azm´any P. s´et´any 1/C, Hungary. [email protected]
Bal´azs Patk´os ∗ MTA–ELTE Geometric and Algebraic Combinatorics Research GroupH–1117 Budapest, P´azm´any P. s´et´any 1/C, Hungary andAlfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences [email protected], [email protected]
Mathematics Subject Classifications: 05C88, 05C89
Abstract
We study the function M ( n, k ) which denotes the number of maximal k -uniformintersecting families F ⊆ (cid:0) [ n ] k (cid:1) . Improving a bound of Balogh, Das, Delcourt, Liu andSharifzadeh on M ( n, k ), we determine the order of magnitude of log M ( n, k ) by provingthat for any fixed k , M ( n, k ) = n Θ( ( kk ) ) holds. Our proof is based on Tuza’s set pairapproach.The main idea is to bound the size of the largest possible point set of a cross-intersecting system. We also introduce and investigate some related functions andparameters. Many problems in extremal combinatorics ask for the maximum possible size that a com-binatorial structure can have provided it satisfies some prescribed property P . Questions ∗ J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences P .) This note is devoted to an applicationof Tuza’s set pair method [17] which provides good bounds for problems of the first typethrough results on problems of the second type.The starting point of Tuza’s method is the following celebrated theorem of Bollob´as. Theorem 1.1 (Bollob´as, [3]) . Let A , A , . . . , A m and B , B , . . . , B m be sets such that | A i | ≤ k and | B i | ≤ l hold for all ≤ i ≤ m . Let furthermore these sets satisfy(1) A i ∩ B i = ∅ for all ≤ i ≤ m ,(2) A i ∩ B j = ∅ for all ≤ i, j ≤ m , i = j .Then P mi =1 1 ( | Ai | + | Bi || Ai | ) ≤ , in particular m ≤ (cid:0) k + ll (cid:1) holds. Pairs satisfying the conditions of Theorem 1.1 will be called cross intersecting set pairs and if we want to emphasize the size condition of the A i ’s and B j ’s, then we call the system( k, l ) -cross intersecting .Modifying Lov´asz’s proof [15] of Theorem 1.1, Frankl [9] and later Kalai [13] obtained thefollowing skew version of the result. Theorem 1.2 (Frankl) . Let A , A , . . . , A m and B , B , . . . , B m be sets such that | A i | ≤ k and | B i | ≤ l , satisfying the conditions(1) A i ∩ B i = ∅ for all ≤ i ≤ m ,(2’) A i ∩ B j = ∅ for all ≤ i < j ≤ m .Still the bound m ≤ (cid:0) k + ll (cid:1) remains valid. Pairs satisfying the conditions of Theorem 1.2 will be called skew cross intersecting setpairs .The vertex set of a (skew) cross intersecting system of set pairs is V = S mi =1 ( A i ∪ B i ). Tuzawas interested in the maximum possible size of the vertex set of a ( k, l )-cross intersectingsystem. Let us write n ( k, l ) = max ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m [ i =1 ( A i ∪ B i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : ( A , B ) , . . . , ( A m , B m ) is a ( k, l )-cross intersecting system ) . Obviously, by Theorem 1.1, we have n ( k, l ) ≤ ( k + l ) (cid:0) k + ll (cid:1) , but the following upper boundwas obtained in [17]. 2 heorem 1.3 (Tuza [17]) . For positive integers k ≤ l we have (cid:18) k + l + 1 k + 1 (cid:19) < n ( k, l ) ≤ k − X i =1 (cid:18) i ⌊ i/ ⌋ (cid:19) + k + l − X i =2 k − (cid:18) il (cid:19) < (cid:18) k + l + 1 k + 1 (cid:19) . Section 2 is devoted to prove another application of the set pair method, the main resultof this note. Apart from antichains the most studied set families are intersecting families.We say that
F ⊆ [ n ] is intersecting if F ∩ F = ∅ holds for all F , F ∈ F . It is well-known that all maximal (unextendable) intersecting families F ⊆ [ n ] have size 2 n − . (Hereand thereafter [ n ] stands for the set { , , . . . , n } .) The investigation of λ ( n ) and Λ( n ),the number of intersecting and maximal intersecting families, respectively, was started in[6]. The exact values are known for small n [5] and determining the order of magnitude oflog λ ( n ) and log Λ( n ) is an easy exercise.Recently, Balogh, Das, Delcourt, Liu, and Sharifzadeh [2] studied the uniform versionof the problem. The famous Erd˝os-Ko-Rado theorem [7] states that an intersecting family F ⊆ (cid:0) [ n ] k (cid:1) can have size at most (cid:0) n − k − (cid:1) if 2 k ≤ n holds. Furthermore, intersecting familiesachieving the extremal size consist of all k -sets containing a fixed element of [ n ] provided2 k < n . Balogh et al. define the function N ( k ) with the property that if n ≥ N ( k ), thenthe number of k -uniform intersecting families is 2 (1+ o (1)) ( n − k − ). In their proof they obtain anupper bound on the number M ( n, k ) of maximal k -uniform intersecting families. Here weimprove on this bound and we determine the order of magnitude of the exponent of n in M ( n, k ) for any fixed k . Theorem 1.4.
For any fixed integer k , as n tends to infinity the function M ( n, k ) satisfies M ( n, k ) = n Θ( ( kk ) ) . Moreover, ≤ lim sup n log M ( n, k ) (cid:0) kk (cid:1) log n ≤ . and lim sup k lim sup n log M ( n, k ) (cid:0) kk (cid:1) log n ≤ holds. The proof of Theorem 1.4 uses the upper bound in Theorem 1.3. In Section 3 we firstprove an upper bound on n ( k, l ) that is weaker than that of Theorem 1.3, but its prooftechnique is completely different: it involves skew cross intersecting systems. Therefore it isnatural to introduce the following analog of the function n ( k, l ): n ( k, l ) = max ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m [ i =1 ( A i ∪ B i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : ( A , B ) , . . . , ( A m , B m ) is a ( k, l )-skew cross intersecting system ) .
3e finish Section 3 by presenting lower and upper bounds on n ( k, l ).Before starting to prove our theorems let us mention that there has been recent activity[4, 10, 11, 12] on the following problem of Balogh, Bohman and Mubay [1] related to maximalintersecting families: let H ( n, k, p ) denote the random k -uniform hypergraph obtained from (cid:0) [ n ] k (cid:1) by keeping each edge with probability p independently of all other choices. What isthe size of the largest intersecting family in H ( n, k, p ) and what its structure looks like.The above mentioned papers settled this question for all interesting values of k = k ( n ) and p = p ( n ). We start with the lower bound of Theorem 1.4. For a family F of sets its covering number τ ( F ) is the minimum size that a transversal G of F can have. A transversal of F is a setmeeting all F ∈ F . Clearly, τ ( F ) ≤ k holds for all intersecting k -uniform families as any setin F is a transversal. Let us define the function f ( k ) by f ( k ) = max {| ∪ F ∈F F | : F is k -uniform intersecting with τ ( F ) = k } . Note that f ( k ) is finite (see [8] ), while the condition τ ( F ) = k is essential in the sensethat | ∪ F ∈F F | could be arbitrarily large if F was k uniform intersecting with τ ( F ) < k .Many similar functions concerning k -uniform intersecting families with covering number k were introduced and studied in [8] (and later by many other researchers). The followingexample is due to Tuza [17]. Construction 2.1.
Let | Y | = 2 k − . For each partition Y as E ∪ E ′ = Y , | E | = | E ′ | = k − we take a new points x , and set E ∪ { x } , E ′ ∪ { x } . In this way we obtain (cid:0) k − k − (cid:1) k -elementsets forming an intersecting family with covering number k , such that the union of these setsconsists of k − (cid:0) k − k − (cid:1) points. Corollary 2.2. (cid:0) kk (cid:1) < k − (cid:0) k − k − (cid:1) ≤ f ( k ) . The following proposition finishes the proof of the lower bound of Theorem 1.4.
Proposition 2.3.
For any positive integers k and n we have (cid:0) nf ( k ) (cid:1) ≤ M ( n, k ) .Proof. Consider a k -uniform intersecting family F with τ ( F ) = k and | ∪ F ∈F F | = f ( k ).As adding more sets to F can only increase the size of the union, we may assume that F ismaximal intersecting. Every set X ∈ (cid:0) [ n ] f ( k ) (cid:1) contains at least one family F X isomorphic to F . As F X = F Y whenever ∪ F ∈F X F = X = Y = ∪ F ∈F Y F , we have at least (cid:0) nf ( k ) (cid:1) differentmaximal intersecting k -uniform subfamilies of (cid:0) [ n ] k (cid:1) .4s we mentioned in the proof, the value of f ( k ) is attained at a maximal intersectingfamily. Note that such a family is unextendable not only by any k -subsets of its underlyingset, but by any k -sets in the universe at all. This kind of maximal intersecting set systemswere studied a lot, the best known upper bound on f ( k ) is due to Majumder [16], statingthat f ( k ) ≤ (1 + o (1)) (cid:0) k − k − (cid:1) .We now turn our attention to the upper bound of Theorem 1.4. We start by describingthe basic ideas of Balogh, Das, Delcourt, Liu, and Sharifzadeh [2]. For a family F ⊆ (cid:0) [ n ] k (cid:1) of sets let I ( F ) = { G ∈ (cid:0) [ n ] k (cid:1) : ∀ F ∈ F : F ∩ G = ∅} , that is if F is intersecting, then I ( F ) denotes the family of those sets that can be added to F preserving the intersectingproperty. Clearly, F is maximal intersecting if and only if F ⊆ I ( F ) holds with equality.For any maximal intersecting family we can assign a subfamily F ⊆ F that is minimalwith respect to the property I ( F ) = F (note that F is not necessarily unique). Then bydefinition, for every F ∈ F there exists a G ∈ I ( F \ { F } ) \ F , thus this G intersects all setsin F but F . Therefore the sets of F and their pairs G satisfy the condition of Theorem 1.1and thus by above, we obtain that |F | ≤ (cid:0) kk (cid:1) . Moreover, if F = { F , F , . . . , F s } and G i is a set in I ( F \ { F i } ) \ F , then the set of pairs { ( A i , B i ) } si =1 with A i = F i , B i = G i for1 ≤ i ≤ s and A i = G i − s , B i = F i − s for s < i ≤ s is skew cross intersecting and thus byTheorem 1.2 the inequality |F | ≤ (cid:0) kk (cid:1) holds. Since the mapping of maximal intersectingfamilies via F 7→ F is injective, Balogh et al. obtained M ( n, k ) ≤ P ( kk ) j =1 (cid:0) ( nk ) j (cid:1) = O ( n k ( kk )).Comparing this to our lower bound, we see that the exponent is off only by a factor of 4 k .In what follows we show how to improve the previous upper bound.In order to obtain our upper bound, we will use the function n ( k, l ). As the argument ofBalogh et al. yields a cross intersecting system in which sets of the first co-ordinate form anintersecting family on their own, we introduce the following: g ( k ) = max {| ∪ si =1 A i | : { ( A i , B i ) } si =1 is ( k, k )-cross intersecting and { A i } si =1 is intersecting } . By definition, we have g ( k ) ≤ n ( k, k ). The following lemma and proposition complete theproof of the upper bound of Theorem 1.4. Lemma 2.4. M ( n, k ) ≤ g ( k ) (cid:0) ng ( k ) (cid:1) .Proof. Let us consider a function f that maps any maximal intersecting k -uniform family F to one of its subfamily F that is minimal with respect to the property that I ( F ) = F .As mentioned earlier, f is injective, F is intersecting and the set of pairs { ( F i , G i ) } |F | i =1 is( k, k )-cross intersecting. Thus by definition | ∪ F ∈F F | ≤ g ( k ) holds. Therefore the setfamilies that can be the image of a maximal intersecting k -uniform family with respect to f are subfamilies of 2 X for some X ∈ (cid:0) [ n ] g ( k ) (cid:1) . The number of such families is not more than2 g ( k ) (cid:0) ng ( k ) (cid:1) . 5hough it was not mentioned in [17], the summation form of the upper bound of Theo-rem 1.3 provides much better estimation in the case k = l . Proposition 2.5.
Let S ( k ) denote Tuza’s upper bound on n ( k, k ) in Theorem 1.3, that is, S ( k ) = P k − i =1 (cid:0) i ⌊ i/ ⌋ (cid:1) . Then(i) g ( k ) ≤ n ( k, k ) ≤ S ( k ) ≤ . · (cid:0) kk (cid:1) ,(ii) s ( k ) := S ( k ) ( kk ) → if k → ∞ . Proof.
Statement (i) can be confirmed easily for k ≤
4, and for k > s ( k ) > s ( k ) is monotone decreasing from k = 4. Moreover the limit cannot be greater than1, since if s ( k ) > (1 + ε ) held with a fixed ε > k , that would imply s ( k +1) s ( k ) ≤ k +34 k +2 1(1+ ε ) ,a contradiction. In the forthcoming section we present lower and upper bounds on n ( k, l ) and n ( k, l ), thatis, on the maximal size of the underlying set of a (skew) cross intersecting system. Construction 3.1 (Erd˝os-Lov´asz, [8]) . Let Y be a set of k − elements. For each subset A ′ ⊂ Y , | A ′ | = k − , we assign a set pair ( A, B ) such that | A | = k = | B | holds, A ′ ⊂ A , ( Y \ A ′ ) ⊂ B and the one element sets A \ Y , B \ Y are disjoint. In this way we obtain (cid:0) k − k − (cid:1) set pairs such that the union of these sets consists of k − (cid:0) k − k − (cid:1) points. This construction slightly improves the general lower bound of Theorem 1.3 on n ( k, l ) inthe special case k = l . Thus in view of Proposition 2.5, this provides Proposition 3.2. k − (cid:0) k − k − (cid:1) ≤ n ( k, k ) ≤ . · (cid:0) kk (cid:1) . In the spirit of Tuza’s approach, the following upper bound is obtained on n ( k, l ). Lemma 3.3. n ( k, l ) ≤ (cid:0) k + ll +1 (cid:1) + (cid:0) k + lk +1 (cid:1) .Proof. Let { ( A i , B i ) } si =1 be a set of cross intersecting pairs with | A i | ≤ k and | B i | ≤ l for all1 ≤ i ≤ s . Let α t = |{ i : | A i \ ( ∪ i − j =1 ( A j ∪ B j )) | ≥ t }| and β t = |{ i : | B i \ ( ∪ i − j =1 ( A j ∪ B j )) | ≥ t }| .Clearly, we have | s [ i =1 ( A i ∪ B i ) | = k X t =1 α t + l X t =1 β t . β t . Observe that if we write B ′ i = B i ∩ ( ∪ i − j =1 ( A j ∪ B j )), then the set of pairs { ( A i , B ′ i ) } si =1 is skew cross intersecting. Moreover | B i \ ( ∪ i − j =1 ( A j ∪ B j )) | ≥ t holds for i if and only if | B ′ i | ≤ l − t . Hence β t is equal to the number of skew crossintersecting set pairs { ( A i , B ′ i ) } where | A i | ≤ k and | B ′ i | ≤ l − t . Applying Theorem 1.2we obtain β t ≤ (cid:0) k + l − tk (cid:1) , and as the role of α t and β t is similar we also have α t ≤ (cid:0) k + l − tl (cid:1) .Consequently, | s [ i =1 ( A i ∪ B i ) | ≤ k X t =1 (cid:18) k + l − tl (cid:19) + l X t =1 (cid:18) k + l − tk (cid:19) = (cid:18) k + ll + 1 (cid:19) + (cid:18) k + lk + 1 (cid:19) . This slightly improves the bound (cid:0) k + l +1 l +1 (cid:1) of Theorem 1.3 when k = l . However, in viewof Proposition 2.5, Tuza’s bound involving a summation is still better even in this case.Recall that n ( k, l ) = max ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m [ i =1 ( A i ∪ B i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : ( A , B ) , . . . , ( A m , B m ) is a ( k, l )-skew cross intersecting system ) . Our second result gives lower and upper bounds on n ( k, l ). In order to do this, we recallwhat a reverse lexicographic order (or sometimes called colex order ) is. Definition 3.4.
A reverse lexicographic order of the k -element subsets of [ n ] is defined bythe relation C < D for
C, D ∈ (cid:0) [ n ] k (cid:1) ⇔ the largest element of the symmetric difference C △ D is in D . Construction 3.5.
Let Y be the set Y = [ k + l ] . Consider the reverse lexicographic orderof all the k -element subsets of Y . Let A i = { a i, , a i, , . . . , a i,k } ( i = 1 . . . (cid:0) k + lk (cid:1) ) be the i th setin this order with the a i,j ’s enumerated in increasing order, and let B i be defined as follows. B i ∩ Y = [ a i,k ] \ A i and let all the sets B i \ Y be pairwise disjoint for all i such that | B i | = k . Proposition 3.6. k + l + (cid:0) k + lk +1 (cid:1) ≤ n ( k, l ) .Proof. Construction 3.5 provides a ( k, l )-skew cross intersecting set system. Indeed, A i ∩ B i = ∅ , while A i ∩ B j = ∅ for i < j , since A i ⊆ [ a i,k ] ⊆ [ a j,k ], hence A i ∩ B j ⊇ A i ∩ ([ a j,k ] \ A j ) = ∅ .Observing that the number of k -sets A j with a j,k = k + c is (cid:0) k + c − k − (cid:1) (assuming c ≤ l ), weget (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m [ i =1 ( A i ∪ B i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k + l + l X c =0 ( l − c ) (cid:18) k + c − k − (cid:19) . l X c =0 ( l − c ) (cid:18) k + c − k − (cid:19) = l − X x =0 x X c =0 (cid:18) k + c − k − (cid:19) = l − X x =0 (cid:18) k + xk (cid:19) = (cid:18) k + lk + 1 (cid:19) , hence the result follows.Note that Construction 3.5 shows that the calculation in Lemma 3.3 to bound β t is tightand thus to obtain better bounds on n ( k, l ) one has to use further ideas.The proof below of the upper bound on n ( k, l ) is based on Tuza’s approach [17] todetermine n ( k, l ). Proposition 3.7.
Let k ≤ l be positive integers. Then n ( k, l ) ≤ (cid:0) k + l +2 k +1 (cid:1) − (cid:0) k + lk (cid:1) − holds.Proof. Let { ( A i , B i ) } si =1 be a set of skew cross intersecting pairs with | A i | ≤ k and | B i | ≤ l for all 1 ≤ i ≤ s and let us define S = [ s ] and M = { ( A i , B i ) } si =1 with A i = A i and B i = B i . If S j and M j are defined for some j ≤ k + l −
2, then let S j +1 ⊂ S j be an indexset minimal with respect to the property that [ i ∈ S j ( A ji ∪ B ji ) = [ i ∈ S j +1 ( A ji ∪ B ji ) . By minimality for every i ∈ S j +1 there exists a point x i ∈ ( A ji ∪ B ji ) \ S l ∈ S j +1 \{ i } ( A jl ∪ B jl ). Letus define A j +1 i = A ji \ { x i } , B j +1 i = B ji \ { x i } for all i ∈ S j +1 and set M j +1 = { ( A j +1 i , B j +1 i ) : i ∈ S j +1 } . Observe that M j is skew intersecting for all 1 ≤ j ≤ k + l − | A ji ∪ B ji | ≤ k + l − j for all i ∈ S j and furthermore s [ i =1 ( A i ∪ B i ) = k + l − X j =1 |M j | . In Tuza’s original proof the M j ’s are cross intersecting and therefore he can use Bollob´as’sinequality to obtain |M j | ≤ (cid:0) k + l − j ⌈ k + l − j ⌉ (cid:1) for any j and |M j | ≤ (cid:0) k + l − jk (cid:1) if j ≤ l − k . AsBollob´as’s inequality is not valid for skew intersecting pairs, therefore we partition M j intosome subsystems indexed by the pairs ( | A i \ A ji | , | B i \ B ji | ). Note that by the constructionof the M j ’s for the index pairs ( a, b ) we have 0 ≤ a, b ≤ j , a + b = j , a ≤ k and b ≤ l . Forsuch a subsystem of M j , indexed by ( a, b ), we can apply Theorem 1.2 and obtain the upperbound (cid:0) k + l − jk − a (cid:1) . Thus adding these up for all M j , j ∈ [1 , k + l − k + l − X j =1 j X a =0 (cid:18) k + l − jk − a (cid:19) = X β ≤ kα ≤ β + l (cid:18) αβ (cid:19) − (cid:18) (cid:19) − (cid:18) k + lk (cid:19) . X β ≤ kα ≤ β + l (cid:18) αβ (cid:19) = k X β =0 l X γ =0 (cid:18) β + γβ (cid:19) = k X β =0 (cid:18) β + l + 1 β + 1 (cid:19) = (cid:18) k + l + 2 k + 1 (cid:19) − , confirming the statement.In [18], Tuza proposed the investigation of the so-called weakly cross intersecting set pairsystems , which are closely related to the cross intersecting set pair systems. Definition 3.8.
Let A , A , . . . , A m and B , B , . . . , B m be sets such that | A i | = k and | B i | = l holds for all ≤ i ≤ m . Let furthermore these sets satisfy(1) A i ∩ B i = ∅ for all ≤ i ≤ m ,(2) A i ∩ B j = ∅ or A j ∩ B i = ∅ for all ≤ i, j ≤ m , i = j .Then the system { ( A i , B i ) } mi =1 is called a ( k, l ) -weakly cross intersecting set pair system.Let m max ( k, l ) denote the largest m ∈ Z for which a ( k, l ) -weakly cross intersecting set pairsystem { ( A i , B i ) } mi =1 exists. Surprisingly, much less is known about the maximum size of a weakly cross intersectingset pair system compared to the original case. Concerning the upper bound, Tuza showed[18] that m max ( k, l ) ≤ ( k + l ) k + l k k l l . Kir´aly, Nagy, P´alv¨olgyi and Visontai gave a construction[14] that provides lim inf k + l →∞ m max ( k, l ) ≥ (2 − o (1)) (cid:0) k + lk (cid:1) . Moreover, they conjectured thelatter result to be sharp: Conjecture 3.9 ([14]) . m max ( k, l ) ≤ (cid:18) k + lk (cid:19) . These questions motivate the investigation of n ( k, l ) = max ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m [ i =1 ( A i ∪ B i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : ( A , B ) , . . . , ( A m , B m ) is a (k,l)-weakly cross intersecting system ) . First, observe that the idea of the proof of Proposition 3.7 works smoothly to obtain anupper bound on n ( k, l ), since we may define weakly cross intersecting set pair systems M j similarly from a given ( k, l )-weakly cross intersecting set pair system. Thus the exact upperbound only depends on m max ( k, l ). Hence, assuming that Conjecture 3.9 holds, we get thedouble of the upper bound of Proposition 3.7.A lower bound follows from 9 onstruction 3.10. Let Y be a set of k + l − elements with k ≤ l . Assign a subset B ′ i ⊂ ( Y \ A ′ i ) of size l − to each k − element subset A ′ i ⊂ Y in such a way that the sets B ′ i are distinct. This can be done due to the K˝onig-Hall theorem and the fact that k ≤ l .For each A ′ i , assign furthermore three distinct elements x i , y i , z i Y . Take the set pairs ( A ′ i ∪ { x i } , B ′ i ∪ { y i } ) , ( A ′ i ∪ { y i } , B ′ i ∪ { z i } ) , ( A ′ i ∪ { z i } , B ′ i ∪ { x i } ) for all i . This way weobtain (cid:0) k + l − k − (cid:1) set pairs such that the union of these sets consists of k + l − (cid:0) k + l − k − (cid:1) points. Proposition 3.11. k + l − (cid:0) k + l − k − (cid:1) ≤ n ( k, l ) . Proof.
The proposition follows from the fact that Construction 3.10 provides a weakly crossintersecting set pair system, which is easy to see.
Acknowledgment.
We would like to thank an anonymous referee for pointing out an errorin a previous version of the manuscript and for their many helpful remarks to improve thepresentation of the paper.
References [1] J. Balogh, T. Bohman, and D. Mubayi. Erd˝os–Ko–Rado in random hypergraphs.
Com-binatorics, Probability and Computing , 18(05):629–646, 2009.[2] J. Balogh, S. Das, M. Delcourt, H. Liu, and M. Sharifzadeh. The typical structure ofintersecting families of discrete structures. arxiv preprint arXiv:1408.2559 , 2014.[3] B. Bollob´as. On generalized graphs.
Acta Mathematica Hungarica , 16(3):447–452, 1965.[4] B. Bollob´as, B. P. Narayanan, and A. M. Raigorodskii. On the stability of the Erd˝os-Ko-Rado theorem. arXiv preprint arXiv:1408.1288 , 2014.[5] A. Brouwer, C. Mills, W. Mills, and A. Verbeek. Counting families of mutually inter-secting sets. the electronic journal of combinatorics , 20(2):P8, 2013.[6] P. Erd˝os. Problems and results in combinatorial analysis.
Proc. Symp. Pure Math.AMS , 19:77–89, 1971.[7] P. Erd˝os, C. Ko, and R. Rado. Intersection theorems for systems of finite sets.
TheQuarterly Journal of Mathematics , 12(1):313–320, 1961.[8] P. Erd˝os and L. Lov´asz. Problems and results on 3-chromatic hypergraphs and somerelated questions.
Infinite and finite sets , 10:609–627, 1975.109] P. Frankl. An extremal problem for two families of sets.
European Journal of Combi-natorics , 3(2):125–127, 1982.[10] M. M. Gauy, H. H`an, and I. C. Oliveira. Erd˝os-Ko-Rado for random hypergraphs:asymptotics and stability. arXiv preprint arXiv:1409.3634 , 2014.[11] A. Hamm and J. Kahn. On Erd˝os-Ko-Rado for random hypergraphs I. arXiv preprintarXiv:1412.5085 , 2014.[12] A. Hamm and J. Kahn. On Erd˝os-Ko-Rado for random hypergraphs II. arXiv preprintarXiv:1406.5793 , 2014.[13] G. Kalai. Intersection patterns of convex sets.
Israel Journal of Mathematics , 48(2-3):161–174, 1984.[14] Z. Kir´aly, Z. L. Nagy, D. P´alv¨olgyi, and M. Visontai. On families of weakly cross-intersecting set-pairs.
Fundamenta Informaticae , 117(1-4):189–198, 2012.[15] L. Lov´asz. Flats in matroids and geometric graphs. In
Combinatorial surveys (Proc.Sixth British Combinatorial Conf., Royal Holloway Coll., Egham, 1977) , pages 45–86.Academic Press London, 1977.[16] K. Majumder. On the maximum number of points in a maximal intersecting family offinite sets. arXiv preprint arXiv:1402.7158 , 2014.[17] Z. Tuza. Critical hypergraphs and intersecting set-pair systems.
Journal of Combina-torial Theory, Series B , 39(2):134–145, 1985.[18] Z. Tuza. Inequalities for two set systems with prescribed intersections.