Multiply intersecting families of sets

Transcription

1 Multiply intersecting families of sets Zoltán Füredi 1 Department of Mathematics, University of Illinois at Urbana-Champaign Urbana, IL61801, USA and Rényi Institute of Mathematics of the Hungarian Academy of Sciences Budapest, P. O. Box 127, Hungary z-furedi@math.uiuc.edu and furedi@renyi.hu and Zsolt Katona Eötvös Loránd University Mailing address: Mexikói út 31/a, Budapest 1145 Hungary zskatona@cs.elte.hu Abstract Let [n] denote the set {1, 2,..., n}, 2 [n] the collection of all subsets of [n] and F 2 [n] be a family. The maximum of F is studied if any r subsets have an at least P s-element intersection and there t s n s are no l subsets containing t + 1 common elements. We show that F i=0 i + t+l s n s t+2 s + l 2 and this bound is asymptotically the best possible as n and t 2s 2, r, l 2 are fixed. AMS classification: 05D05; 05B25 Keywords: extremal problems for families of finite sets 1 Research supported in part by the Hungarian National Science Foundation under grants OTKA T , T and by the National Science Foundation under grant DMS

2 Z. Füredi & Zs. Katona: Multiply intersecting families 2 1 Intersecting families: cylinders and Hamming balls Let [n] denote the set {1, 2,..., n} and 2 [n] the collection of all subsets of [n], and [n] k is the set of k-element subsets of [n]. A simple theorem of Erdős, Ko and Rado [6] says that the maximum of F is 2 n 1 if every two members of a family F 2 [n] have a non-empty intersection. Such an F is called an intersecting family. A maximum intersecting family M can be obtained by considering all the subsets containing one fixed element. This M sometimes is called a cylinder, or trivially intersecting, since M =. The above theorem was extended by Gy. Katona [13] as follows. If a a set system F on [n] is s-intersecting, i.e., any two sets in the system have intersection of size at least s, then for n + s is even F Bn, 1 2 n + s where Bn, x := {A [n] : A x}, a Hamming ball. For the case n + s is odd one has F 2 Bn 1, 1 2 n + s 1, and the optimal family is a combination of a cylinder and a Hamming ball. This is often the case in the theory of intersection families, especially if one considers uniform families where all sets have the same sizes, see, e.g., Ahlswede and Khachatrian s [1] solution for the Erdős Frankl conjecture. In general, we say that F has the Ir, s property also called r-wise s-intersecting if F 1 F 2... F r s holds for every F 1, F 2,..., F r F, 1 and let fn; Ir, s denote the size of the largest r-wise s-intersecting family on n elements. Taking all subsets containing a fixed s-element set i.e., a cylinder shows that fn; Ir, s 2 n s holds for all n s 0. One of the nicest results of the field is due to Frankl [7] that fn; Ir, s = 2 n s holds if and only if n < r + s or s 2 r r 1 with the possible but unlikely exception of the case r, s = 3, 4. An excellent survey of these families is due to Frankl [8]. 2 Intersecting families: codes and packings If we have an upper bound on the intersection sizes, then the extremal families are codes, designs, and packings. More precisely, we say that F has the Il, t property if F 1 F 2... F l t holds for every distinct F 1, F 2,..., F l F, 2 and let fn; Il, t denote the size of the largest family satisfying Il, t on n elements. A family of k-subsets of [n] with the Il, t property is called an n, k, t+1, l 1 packing and its maximum

3 Z. Füredi & Zs. Katona: Multiply intersecting families 3 size is denoted by P n, k, t + 1, l 1. More generally, a family F 2 [n] is called an n, k +, j, λ packing if F k holds for every F F and F[X] λ for every j-subset X [n]. Here, as usual, F[X] denotes the family {F : X F F}, F[x] stands for F[{x}], and deg F x := F[x]. The maximum size of such a packing is denoted by P n, k +, j, λ. Simple double counting gives that P n, k, j, λ P n, k +, j, λ λ k. 3 j If here equalities hold for F [n] k, then it is called an Sλ n, k, j block design, and in the case λ = 1, a Steiner system. For more details about packings see [3, 4]. The existence of designs and the determination of the packing number is a very difficult question, here we only recall a folklore result about the case j = k 1. λ n k n P n, k, k 1, λ k n k 1 n j P n, k +, k 1, λ λ k n. 4 k 1 Proof: The upper bound follows from 3. To show the lower bound we give a construction F. For every x [n] consider F x := {F [n] k : f F f x mod n}. It satisfies the property that every k 1-element set is contained in at most one member of F x, so F 1 F 2 F n is a decomposition of the complete hypergraph [n] k into n, k, k 1, 1 packings. Define F as the union of the λ largest ones among these F x s. Then F λ n n k = λ k n k n n k 1. One can see that the determination of fn; Il, t is equivalent to the determination of the packing function estimated in 4. Proposition 1 Suppose that l 2, t 1 are integers. Then fn; Il, t = n + P n, t + 2 +, t + 1, l 2. i 0 i t+1 Proof: Easy. Let F 2 [n] be a set system with property Il, t. We may suppose that all subsets of [n] with at most t elements belong to F because if X t, then F {X} has the Il, t property. We also show that we may suppose that all t + 1-element subsets belong to F. Indeed, if X [n] t+1, X / F and F[X] l 2, then again F {X} also has the Il, t property. Finally,

4 Z. Füredi & Zs. Katona: Multiply intersecting families 4 if F[X] = l 1 > 0, then take a member F F[X] and replace it by X. The obtained new family F \ {F } {X} has the Il, t property, too. One can repeat this process until all members of [n] t+1 belong to F. Then, obviously, the family F \ 0 i t+1 [n] i is an n; t + 2 +, t + 1, l 2 packing. Note that the above proof implies the following slightly stronger result. If F has the Il, t property on [n] and each F F has size at least t + 1, then n F + P n, t + 2 +, t + 1, l 2. 5 t + 1 Conjecture 2 Suppose that k, j and λ are positive integers. Then for sufficiently large n, n > n 0 := n 0 k, j, λ the packing functions in 4 have the same values P n, k, j, λ = P n, k +, j, λ. 3 Simultaneous restrictions The Problem. Determine fn; Ir, s, Il, t, the maximal size of F 2 [n] satisfying both the conditions 1 and 2. Although there were several partial results and we will cite some of those later this problem was proposed in this generality only in [14]. We suppose that r, l 2, although the cases r = 1 and l = 1 are also interesting. It was also proved in [14] that the following asymptotic holds for fixed l as n fn; I2, 1, Il, 1 = l 1n + on. Our first result is a simple proof for a more precise version of this. Theorem 3 If any r 2 members of F 2 [n] have a non-empty intersection but the intersection of any l distinct members contains at most one element, then F l 1n, i.e., fn; Ir, 1, Il, 1 l 1n holds for every n and r, l 2.

5 Z. Füredi & Zs. Katona: Multiply intersecting families 5 The proof is postponed to Section 4. There we also discuss related results, linear and almost linear hypergraphs, a topic started by de Bruijn and Erdős [5], who proved the case l = 2 of Theorem 3. In Section 5 we give an asymptotic solution of the general problem for the case t 2s. More exactly, we reduce it to a packing function discussed in 4. We show that Theorem 4 Suppose that t 2s 2, r 2, l 2 and n > n 0 := n 0 r, s, l, t. Then for r 3 fn; Ir, s, Il, t = fn s; Il, t s. If r = 2 then fn; I2, s, Il, t fn s; Il, t s + l 2, and here equality holds if Conjecture 2 holds for n s, t + 2 s,, l 2-packings and s l 1. Then Proposition 1 and 4 imply for r 3 that n s f = + P n s; t + 2 s +,, l 2 i i=0 = 1 + o1 1 + l 2 n, t + 2 s and the same asymptotic result holds for r = 2. 4 The Case s = t = 1 Here we prove Theorem 3. The following is a slight generalization of a result of Motzkin [15] who proved the case c = 1. In fact, the case c = 1 implies the lemma for all real c > 0. Lemma 1 Let G = GA, B; E be a bipartite graph with E, and let c be a positive real number. Suppose that no vertex in A is adjacent to all vertices in B and that for every pair of non-joined vertices a A, b B, a, b E one has dega c degb. Then A c B. Proof: Let p and q be positive integers satisfying p/q c. We will show p/q A B, and since p/q c could be arbitrarily small and since G is finite this will suffice to show c A B.

6 Z. Füredi & Zs. Katona: Multiply intersecting families 6 Let G be a bipartite graph with vertices A B such that A is p copies of A, and B is q copies of B and if a A is a copy of a A and b B is a copy of b B then a is joined to b in G if and only if a is joined to b in G. Then in case of a, b / E, p deg G a = q deg G a q q deg Gb = p deg G b = deg G b. Thus the conditions of the Motzkin s lemma the case c = 1 hold for G. This implies A B. Hence p q A B holds, as desired. Proof of Theorem 3. Let F be an intersecting family, with no l sets containing two common elements. Define a bipartite graph GA, B; E as follows. A := [n], B := F and x [n] is adjacent to F F if x F. If this graph has no edge then F = for every F F, implying F 1. If there is an a A which is adjacent to all vertices of B, then the element a is contained in every member of F. However there are at most l 1 members of F containing a pair of [n], i.e., F[x, y] l 1, 6 where F[x, y] := {X F : x, y X}. Using this inequality for the pairs a, y, we get that every element y a is contained in at most l 1 sets of F. Thus F 1 + l 1n 1 l 1n. We will show that in any other case the main constraint of Lemma 1 holds with c = l 1 which is positive, since l 2. Let x [n] = A and F F = B be a non adjacent pair in G, i.e., x / F. Consider F[x] := {X F : x X}. We have F[x] = deg G x. Since F is intersecting, all members of F[x] intersect F, thus F[x] = y F F[x, y]. Then 6 implies F[x] l 1 F. This means deg G x l 1 deg G F. Hence Lemma 1 can be applied to G giving l 1n = l 1 A B = F. 5 Proof of Theorem 4 Let Σ := i=0 n s i + P n s, t + 2 s +,, l 2. One can give a family of Σ sets which have s common elements and satisfies the intersection properties. Let P be a packing with P n s, t + 2 s +,, l 2 members on the underlying

7 Z. Füredi & Zs. Katona: Multiply intersecting families 7 set [n] \ [s]. Let P = {P [s] : P P }. Let F 0 = {F [n] : F t + 1; [s] F } P. Hence fn; Ir, s, Il, t F 0 = Σ. 7 For r = 2 one can construct a larger family. Suppose that there is an optimal n s, t + 2 s +, t + 1 s, l 2 packing consisting of only t + 2 s-element sets, i.e. Conjecture 2 holds for these values. Then remove [s] from F 0 and add min{s, l 1} sets of the form [n] \ {i}, 1 i s. One can easily see that the obtained family F 1 is pairwise s-intersecting and satisfies the Il, t property. Let F be a family with the properties 1 and 2. We have to show the other inequality F Σ for r 3, and F Σ + l 2 for r = 2. First, we discuss the case F s. This means that all the members of F have s common elements, say, [s] F for every F F. Consider F = {F \ [s] : F F}. In F the intersection of any l members has at most t s elements, it satisfies the property Il, t s. So every elements are contained in at most l 1 members of F. Hence F = F Σ by Proposition 1. From now on, we suppose that F < s. Define Fi := {F F : F = i} and F j = Fj Fj We have F = F1 F2.... Trivially Fi = if i < s. The family Fi is i-uniform on [n] vertices and satisfies I2, s. So Erdős-Ko-Rado theorem [6] says that n s Fi for n > ni, s. 8 i s We use this estimate if s i t. By 7 and 4 we can suppose that F t s i=0 implies F t l 2 t + 2 s We need an upper bound on the number of large sets. n s i l 2 t+2 s n t 2 n s n s n t 2 n s n s. Hence 8 9 Lemma 2 If A F s for every F F then F k A s k s n s l 1. Proof: Let A = {X [n] t+1 : A X s}. Clearly A A s n s. Every t + 1 elements are contained in at most l 1 members of F, and every F F k contains at least k s members of A. Hence F k k s A l 1.

8 Z. Füredi & Zs. Katona: Multiply intersecting families 8 Since t 2s 2 the fraction k s / k s is arbitrarily small for all sufficiently large k k0. For example, for k > k 0 t, s := 4s 2 + 6t 2 we get k s k s < 2t k < 1 3t. Split F t + 1 into two parts, G F k 0, i.e., G := Ft + 1 Ft Fk 0 1. We are going to give an upper bound on the size of the family F k 0. If it is non-empty, then choose F 0 F k 0 with the minimum size among its members, i.e., F F 0 for every F F k 0. Denote the size of F 0 by f 0. We can use F 0 as A in Lemma 2. We obtain f0s F k 0 = F f 0 f0 s n s l 1 < l 1 n s. 3t Comparing this to 9 we get for n > k 0 that G = F t + 1 F f l 2 n t 2 l 1 n s t + 2 s n s 3t l 1 n s. 10 3t Proposition 5 G = s. Proof: If G s 1, then there is a set A [n], A < 3k 0 such that A G s + 1 for every G G. Indeed, either there is an A G meeting all other members of G in at least s + 1 elements, or we can find G 1, G 2 G with G 1 G 2 = s. Then there exists a G 3 G not containing G 1 G 2. Thus G 1 G 2 G 3 s 1. Then G 1 G 2 G 3 is suitable for A. The existence of such an A in the case G s + 1 is obvious. The Il, t property implies that A n s 1 G s + 1 t s This contradicts 10 for n > n 0 k, s. l 1 3k0 s + 1 n s n s l 1. Thus, we may assume that [s] G. Let S := {F F t + 1 : [s] F } and let H = {F F t + 1 : [s] F }. We have F t + 1 = S H. The family S := {F \ [s] : F S} has the

9 Z. Füredi & Zs. Katona: Multiply intersecting families 9 Il, t s property on n s elements. Moreover each member of S has size at least. So Proposition 1, more exactly 5, implies that n s S = S + P n s, t + 2 s +,, l If H = then S = F t + 1 and 11 and 8 imply F Σ, and we are done. From now on we suppose that H. Let H 1 be a minimal size member in H, H 1 = h. To estimate S consider the family C := {C [n] : C [s], C = t + 1}. Since Ft + 1 G S we have that Ft + 1 C. Every member of S t + 2 contains at least t + 2 s members of C. On the other hand, every member of C is contained in at most l 1 members of F. We obtain, that n s t + 2 s S Ft Ft + 1 l 1 C = l 1. Rearranging we get S 1 + l 2 n s C Ft + 1. t + 2 s t + 2 s For F Ft + 1, F \ [s] can not be contained in [n] \ H 1. Hence C Ft + 1 n s h. Also the fraction t s + 1/t s + 2 is at least 2/3. We obtain S 1 + l 2 n s 2 t + 2 s 3 n h s. 12 To give upper bound to H we use Lemma 2 with an arbitrary A G. Since A k 0 and H h for every H H, we have k0 H s n s h s l 1 < 2t n s l h Adding up the upper bounds 12 and 13 and comparing to the lower bound 9 we get the following. 1 + l 2 t + 2 s n t 2 n s F t + 1 = S + H n s n s 2 n h s + 2t 3 h 1 + l 2 t + 2 s n s l 1. Rearranging we obtain 2 n h s 2t n s + 1 n s. 14 3l 1 h n s

10 Z. Füredi & Zs. Katona: Multiply intersecting families 10 We can redefine k 0 t, s as k 0 t, s, l, sufficiently large depending only on t, s, and l, and suppose that n is sufficiently large compared to this new k 0, i.e., n > n 0 t, s, l. Then 14 implies h > l 1 l n + t. Hence the Il, t property implies that H l 1. Again 11 and 8 imply F Σ + H Σ + l 1. Since F Σ, we get from 8 that Fs + 1 n s 1 H n s l 1 > k0 > s + 2. The members of Fs + 1 pairwise meet in s elements so either Fs + 1 s + 2 and then Fs + 1 s+2 s+1, or Fs + 1 = s. Let S 0 be the s-set contained in every F Fs + 1. If a set F meets every member of Fs + 1 in at least s elements and F < Fs + 1, then S 0 F. This implies that G S 0 and also that S 0 is contained in every member of Fi for s i t, too. Then Proposition 5 implies that S 0 = [s]. We also get that holds for every H H. H [s] = s 1 and H Fs + 1 \ [s] Consider F 1 F 2 Fs + 1, and H 1 H. Then F 1 F 2 H 1 s 1. This is a contradiction for r 3 hence F Σ in this case. Finally, suppose that r = 2. Since H the set system F cannot contain [s], so Fs = 0 and thus F \ H fn s, Il, t s 1. Thus F Σ 1 + l 1. 6 Multihypergraphs In the previous theorems we did not allow multiplied sets. Consider a sequence of sets F = {F 1, F 2,..., F m } of subsets of [n] with properties Ir, s and Il, t where now repetition is allowed. If we can have multi-sets with size s F t then this sequence can be arbitrarily long. Define f n; Ir, s, Il, t as max m where F is a family of multi-subsets of [n] satisfying the intersection conditions 1, 2 with all members having at least t + 1 elements. Theorem 6 Suppose that t 2s 2, r 2, l 2 and n > n 0 := n 0 r, s, l, t. Then for r 3 n s f n; Ir, s, Il, t = f n s; Il, t s = l 1

11 Z. Füredi & Zs. Katona: Multiply intersecting families 11 If r = 2 then f n; I2, s, Il, t = f n s; Il, t s + l 1 n s = l 1 + l 1. The proof of the upper bound is nearly the same as of Theorem 4. The packing problem giving the lower bound is trivial here, the extremal family consists of only t + 1-element sets l 1 copies each for r 3 and some n 1-element sets in the case r = 2. The analog of Theorem 3 holds. Theorem 7 f n; I2, 1, Il, 1 = l 1n holds for every n and l 2. One can take multiple copies of the very same extremal configurations as in [5] and [15], namely l 1 copies of the lines of a finite projective plane if such exists, so in this case n = q 2 + q + 1 or l 1 copies of n 1 pairs through an element x and the set [n] \ {x} for all n. 7 Conclusion It was not unknown in the literature to investigate intersecting families of sets with upper and lower bounds on the intersection sizes. For example it was conjectured in [9] that using our notation fn; I2, 1, I2, k = n 1. i 0 i k Taking all the at most k + 1-element sets containing a given element shows that this is, indeed, the best possible. This was proved for n > 100k 2 / logk + 1 by Frankl and Füredi [9] using the so-called -system method, for n 2k + 2 and for 6k + 1 n 1/5k by Pyber [16] using the permutation method in an ingenious way. Finally, Ramanan [17] proved the conjecture for all n without characterizing the extremal families using the method of multilinear polynomials, building on earlier successes by among others Alon, Babai and Suzuki [2]. A second proof was given based on the same technique by Sankar and Vishwanathan [18].

12 Z. Füredi & Zs. Katona: Multiply intersecting families 12 In the present paper we extended those results for multiple intersections whenever n is large. There is a renewed interest to multiple intersection problems, see, e.g., Grolmusz [10] and Grolmusz and Sudakov [11] for recent developments. 8 Acknowledgements The authors are greatful to the referees for pointing out an error in an earlier version of this paper. References [1] R. Ahlswede and L. H. Khachatrian, A pushing-pulling method: new proofs of intersection theorems, Combinatorica , [2] N. Alon, L. Babai and H. Suzuki, Multilinear polynomials and Frankl Ray-Chaudhuri Wilson type intersection theorems, J. Combin. Theory, Ser. A , k [3] A. E. Brouwer, Packing and covering of t -sets, in: Packing and Covering in Combinatorics, edited by A. Schrijver, Mathematical Centre Tracts 106, Matematisch Centrum, Amsterdam, pp [4] A. E. Brouwer, Block Designs, Chapter 14 in: Handbook of Combinatorics, R. Graham, M. Grötschel and L. Lovász Eds., Elsevier, Amsterdam, [5] N. G. de Bruijn and P. Erdős, On a combinatorial problem, Nederl. Akad. Wetensch., Proc , = Indagationes Math , [6] P. Erdős, Chao Ko, and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser , [7] P. Frankl, Multiply-intersecting families, J. Combin. Theory, Ser. B , [8] P. Frankl, Extremal set systems, in: Handbook of Combinatorics, Vol. 1, 2, pp , R. Graham, M. Grötschel and L. Lovász Eds., Elsevier, Amsterdam, [9] P. Frankl and Z. Füredi, Families of finite sets with missing intersections, in: Finite and infinite sets, Vol. I, II Eger, 1981, pp , Colloq. Math. Soc. János Bolyai, 37, North-Holland, Amsterdam, [10] V. Grolmusz, Set-systems with restricted multiple intersections and explicit Ramsey hypergraphs, Electronic J. of Combinatorics, , R8. [11] V. Grolmusz and B. Sudakov, On k-wise set-intersections and k-wise hamming-distances, J. Combin. Theory, Ser. A , [12] T. Gustavsson, Decompositions of large graphs and digraphs with high minimum degree, Doctoral Dissertation, Dept. of Mathematics, Univ. of Stockholm, [13] Gy. Katona, Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hungar , [14] Zsolt Katona, Intersecting families of sets, no l containing two common elements, Discrete Math , [15] Th. Motzkin, The lines and planes connecting the points of a finite set, Trans. Amer. Math. Soc , [16] L. Pyber, An extension of a Frankl-Füredi theorem, Discrete Math ,

13 Z. Füredi & Zs. Katona: Multiply intersecting families 13 [17] G. V. Ramanan, Proof of a conjecture of Frankl and Füredi, J. Combin. Theory, Ser. A , [18] Arvind Sankar and Sundar Vishwanathan, Multilinear polynomials and a conjecture of Frankl and Füredi, J. Combin. Theory, Ser. A ,