On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method

We study the function $M(n,k)$ which denotes the number of maximal $k$-uniform intersecting families $F\subseteq \binom{[n]}{k}$. Improving a bound of Balogh at al. on $M(n,k)$, we determine the order of magnitude of $\log M(n,k)$ by proving that for any fixed $k$, $M(n,k) =n^{\Theta(\binom{2k}{k})}$ holds. Our proof is based on Tuza's set pair approach. 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 and parameters.


Introduction
Many problems in extremal combinatorics ask for the maximum possible size that a combinatorial structure can have provided it satisfies some prescribed property P . Questions about the size of the 'underlying set' of the combinatorial structure are much less frequently asked. (In many cases, this size is part of property P .) This note is devoted to an application of Tuza's set pair method [17] which provides good bounds for problems of the first type through results on problems of the second type. The starting point of Tuza's method is the following celebrated theorem of Bollobás.
Theorem 1.1 (Bollobás, [3]). Let A 1 , A 2 , . . . , A m and B 1 , B 2 , . . . , B m be sets such that |A i | ≤ k and |B i | ≤ l hold for all 1 ≤ i ≤ m. Let furthermore these sets satisfy 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 Lovász's proof [15] of Theorem 1.1, Frankl [9] and later Kalai [13] obtained the following skew version of the result. Theorem 1.2 (Frankl). Let A 1 , A 2 , . . . , A m and B 1 , B 2 , . . . , B m be sets such that |A i | ≤ k and |B i | ≤ l, satisfying the conditions Still the bound m ≤ k+l l remains valid.
Pairs satisfying the conditions of Theorem 1.2 will be called skew cross intersecting set pairs.
The vertex set of a (skew) cross intersecting system of set pairs is V = m i=1 (A i ∪B i ). Tuza was interested in the maximum possible size of the vertex set of a (k, l)-cross intersecting system. Let us write Obviously, by Theorem 1.1, we have n(k, l) ≤ (k + l) k+l l , but the following upper bound was obtained in [17]. [17]). For positive integers k ≤ l we have Section 2 is devoted to prove another application of the set pair method, the main result of this note. Apart from antichains the most studied set families are intersecting families. We say that F ⊆ 2 [n] is intersecting if F 1 ∩ F 2 = ∅ holds for all F 1 , F 2 ∈ F . It is wellknown that all maximal (unextendable) intersecting families F ⊆ 2 [n] have size 2 n−1 . (Here and thereafter [n] stands for the set {1, 2, . . . , 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 of log λ(n) and log Λ(n) is an easy exercise.
The proof of Theorem 1.4 uses the upper bound in Theorem 1.3. In Section 3 we first prove an upper bound on n(k, l) that is weaker than that of Theorem 1.3, but its proof technique is completely different: it involves skew cross intersecting systems. Therefore it is natural to introduce the following analog of the function n(k, l): m , B m ) is a (k, l)-skew cross intersecting system . We finish Section 3 by presenting lower and upper bounds on n 1 (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 maximal intersecting families: let H(n, k, p) denote the random k-uniform hypergraph obtained from

Proof of the main theorem
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 set meeting all F ∈ F . Clearly, τ (F ) ≤ k holds for all intersecting k-uniform families as any set in F is a transversal. Let us define the function f (k) by Note that f (k) is finite (see [8] ), while the condition τ (F ) = k is essential in the sense that | ∪ 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 following example is due to Tuza [17].
we take a new points x, and set E ∪ {x}, E ′ ∪ {x}. In this way we obtain 2k−2 k−1 k-element sets forming an intersecting family with covering number k, such that the union of these sets consists of 2k − 2 + 1 2 2k−2 k−1 points.
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 n f (k) ≤ 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 is maximal intersecting. Every set X ∈ [n] f (k) contains at least one family F X isomorphic to k .
As we mentioned in the proof, the value of f (k) is attained at a maximal intersecting family. Note that such a family is unextendable not only by any k-subsets of its underlying set, but by any k-sets in the universe at all. This kind of maximal intersecting set systems were studied a lot, the best known upper bound on f (k) is due to Majumder [16], stating We now turn our attention to the upper bound of Theorem 1.4. We start by describing the basic ideas of Balogh, Das, Delcourt, Liu, and Sharifzadeh [2]. For a family F ⊆ [n] denotes the family of those sets that can be added to F preserving the intersecting property. 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 0 ⊆ F that is minimal with respect to the property I(F 0 ) = F (note that F 0 is not necessarily unique). Then by definition, for every F ∈ F 0 there exists a G ∈ I(F 0 \ {F }) \ F , thus this G intersects all sets in F but F . Therefore the sets of F 0 and their pairs G satisfy the condition of Theorem 1.1 and thus by above, we obtain that ). Comparing this to our lower bound, we see that the exponent is off only by a factor of 4k. 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 of Balogh et al. yields a cross intersecting system in which sets of the first co-ordinate form an intersecting family on their own, we introduce the following: By definition, we have g(k) ≤ n(k, k). The following lemma and proposition complete the proof of the upper bound of Theorem 1.4.
Lemma 2.4. M(n, k) ≤ 2 2 g(k) n g(k) . Proof. Let us consider a function f that maps any maximal intersecting k-uniform family F to one of its subfamily F 0 that is minimal with respect to the property that I(F 0 ) = F . As mentioned earlier, f is injective, F 0 is intersecting and the set of pairs holds. Therefore the set families that can be the image of a maximal intersecting k-uniform family with respect to f are subfamilies of 2 X for some X ∈ [n] g(k) . The number of such families is not more than 2 2 g(k) n g(k) .
Though it was not mentioned in [17], the summation form of the upper bound of Theorem 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, Proof. Statement (i) can be confirmed easily for k ≤ 4, and for k > 4 simple inductive argument works. For statement (ii), one can easily check that s(k) > 1 holds, and the sequence s(k) is monotone decreasing from k = 4. Moreover the limit cannot be greater than 1, since if s(k) > (1+ε) held with a fixed ε > 0 for all k, that would imply s(k+1) s(k) ≤ 4k+3 4k+2 1 (1+ε) , a contradiction.

Bounds on the size of the vertex set
In the forthcoming section we present lower and upper bounds on n(k, l) and n 1 (k, l), that is, on the maximal size of the underlying set of a (skew) cross intersecting system.
In the spirit of Tuza's approach, the following upper bound is obtained on n(k, l).
Hence β t is equal to the number of skew cross intersecting set pairs Applying Theorem 1.2 we obtain β t ≤ k+l−t k , and as the role of α t and β t is similar we also have α t ≤ k+l−t l . Consequently, This slightly improves the bound k+l+1 l+1 of Theorem 1.3 when k = l. However, in view of Proposition 2.5, Tuza's bound involving a summation is still better even in this case.
Our second result gives lower and upper bounds on n 1 (k, l). In order to do this, we recall what 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 by the relation C < D for C, D ∈ [n] k ⇔ 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 order of all the k-element subsets of Y . Let A i = {a i,1 , a i,2 , . . . , a i,k } (i = 1 . . . k+l k ) be the ith set in this order with the a i,j 's enumerated in increasing order, and let B i be defined as follows.
Proposition 3.6. k + l + k+l k+1 ≤ n 1 (k, l). Proof. Construction 3.5 provides a (k, l)-skew cross intersecting set system. Indeed, Observing that the number of k-sets A j with a j,k = k + c is k+c−1 hence the result follows.
Note that Construction 3.5 shows that the calculation in Lemma 3.3 to bound β t is tight and thus to obtain better bounds on n 1 (k, l) one has to use further ideas.
The proof below of the upper bound on n 1 (k, l) is based on Tuza's approach [17] to determine n(k, l).
If S j and M j are defined for some j ≤ k + l − 2, then let S j+1 ⊂ S j be an index set minimal with respect to the property that By minimality for every i ∈ S j+1 there exists a point In Tuza's original proof the M j 's are cross intersecting and therefore he can use Bollobás's inequality to obtain |M j | ≤ k+l−j ⌈ k+l−j 2 ⌉ for any j and |M j | ≤ k+l−j k if j ≤ l − k. As Bollobás's inequality is not valid for skew intersecting pairs, therefore we partition M j into some subsystems indexed by the pairs (|A i \ A j i |, |B i \ B j i |). Note that by the construction of the M j 's for the index pairs (a, b) we have 0 ≤ a, b ≤ j, a + b = j, a ≤ k and b ≤ l. For such a subsystem of M j , indexed by (a, b), we can apply Theorem 1.2 and obtain the upper bound k+l−j k−a . Thus adding these up for all M j , j ∈ [1, k + l − 1], we get Here β≤k α≤β+l confirming the statement.
In [18], Tuza proposed the investigation of the so-called weakly cross intersecting set pair systems, which are closely related to the cross intersecting set pair systems.
Definition 3.8. Let A 1 , A 2 , . . . , A m and B 1 , B 2 , . . . , B m be sets such that |A i | = k and |B i | = l holds for all 1 ≤ i ≤ m. Let furthermore these sets satisfy 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 pair exists. Surprisingly, much less is known about the maximum size of a weakly cross intersecting set 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 . Király, Nagy, Pálvölgyi and Visontai gave a construction [14] that provides lim inf k+l→∞ m max (k, l) ≥ (2 − o(1)) k+l k . Moreover, they conjectured the latter result to be sharp: First, observe that the idea of the proof of Proposition 3.7 works smoothly to obtain an upper bound on n 2 (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 upper bound only depends on m max (k, l). Hence, assuming that Conjecture 3.9 holds, we get the double of the upper bound of Proposition 3.7.
A lower bound follows from Construction 3.10. Let Y be a set of k + l − 1 elements with k ≤ l. Assign a subset B ′ i ⊂ (Y \ A ′ i ) of size l − 1 to each k − 1 element subset A ′ i ⊂ Y in such a way that the sets B ′ i are distinct. This can be done due to the Kőnig-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 for all i. This way we obtain 3 k+l−1 k−1 set pairs such that the union of these sets consists of k + l − 1 + 3 k+l−1 k−1 points.
Proof. The proposition follows from the fact that Construction 3.10 provides a weakly cross intersecting set pair system, which is easy to see.