The Intersection Theorem for Direct Products

Transcription

1 Europ. J. Combinatorics , Article No. ej9803 The Intersection Theorem for Direct Products R. AHLSWEDE, H.AYDINIAN AND L. H. KHACHATRIAN c 1998 Academic Press 1. INTRODUCTION Before we state the intersection problem for direct products and our solution, we set up our notation and give a setch of some ey steps in the extremal theory of set intersections. N denotes the set of positive integers and for i, j N, i < j, the set i, i + 1,..., j} is abbreviated as [i, j]. For, n N, n, weset =F : F [1, n]}, =F : F =}. Similarly, for a finite set we use and. A system of sets A is called t-intersecting, if A 1 A t for all A 1, A A, and I n, t denotes the set of all such systems. We denote by I n,, t the set of all -uniform t-intersecting systems, that is, } I n,, t = A I n, t : A. The investigation of the function Mn,, t = max A, 1 t n, A I n,,t and the structure of maximal systems was initiated by Erdös, Ko, and Rado [6]. THEOREM 1.1 [6]. For 1 t and n n 0, t suitable n t Mn,, t =. t The smallest n 0, t has been determined by Franl [8] for t 15 and subsequently by Wilson [15] for all t: n 0, t = t + 1t + 1. In the recent paper [1] all the remaining cases t < n < t + 1t /98/ $30.00/0 c 1998 Academic Press

2 650 R. Ahlswede et al. have been settled by proving the General Conjecture of Franl [8], which stated that for 1 t n Mn,, t = max 0 i n t F i, where } F i = F : F [1, t + i] t + i, 0 i n t. 1.1 THEOREM 1. [1]. For 1 t n with i t r+1 t 1 < n < t t 1 r for some r N 0}, we have Mn,, t = F r and F r is up to permutations the unique optimum by convention t 1 r r = 0. ii t r+1 t 1 = n for r N 0} we have = for Mn,, t = F r = F r+1 and an optimal system equals up to permutations either F r or F r+1. A very special case of Theorem 1. establishes the validity of the long-standing so-called 4m-conjecture see [7, p. 56] and survey [5]. In connection with Theorem 1. we note that, using the ideas of [1], in [] maximal nontrivial intersecting systems see [1] have been determined completely, and in [3] the problem of optimal anticodes in Hamming spaces has been solved. The following problem, initiated by Franl, arose in connection with a result of Sali [14]. Let n = n 1 + +n m, = m and = 1 m with i =n i. Define H = F : F i = i for i = 1,...,m For given integers t i, 1 t i i, 1 i m, we say that A H is t 1,...,t m - interesting, if for every A, B A there exists an i, 1 i m, such that A B i t i holds. Denote the set of all such systems by I H, t 1,...,t m. The problem is to determine MH, t 1,...,t m = max A. A I H,t 1,...,t m Later, instead of I H, t 1,...,t m resp. MH, t 1,...,t m, we use the abbreviations I H resp. MH. The case t 1 = t = =t m = 1 has been solved by Franl [10]. THEOREM 1.3 [10]. Let m nm 1 n 1 1 and t 1 = t = =t m = 1, then MH = 1 n 1 H. }.

3 Intersection theorem for direct products 651 The proof is based on the eigenvalue method the idea of which is due to A. J. Hoffman see [11] and developed by Lovász [13]. In the same paper [10] the following more general result has been stated without proof. THEOREM 1.4 [10]. Let the integers n i, i, t i satisfy n i i = 1,...,m, then ni t i i t i + 1t i + 1 for MH = max i i t i H. ni i In the present paper we determine MH for all parameters. Our result is THEOREM 1.5. Let n i i t i 1 for i = 1,...,m, then MH = max i Mn i, i, t i H. We emphasize that the combination of this Theorem and Theorem 1. gives an explicit value of MH. The proof of the Theorem is purely combinatorial and heavily but not only! based on ideas and methods from [1]. An essential ingredient is a result from [4]. REMARKS. 1 We can always assume that n i > i t i for all i = 1,...,m, because otherwise obviously MH = H. With the set H, having parameters n i i t i, n i > i t i, we consider any twin set H 1 m = 1 m, where either 1 = i or i = n i i and the intersection numbers are t i = t i,if i = i, and t i = n i i + t i,if i = n i i. Clearly MH = MH holds. ni i. LEFT COMPRESSED SETS, GENERATING SETS AND THEIR PROPERTIES We recall first some well-nown and also more recent concepts, which can be found in [1]. Then we give extensions to direct products and basic properties of generating sets. DEFINITION.1. Let B 1 = i 1,...,i }, i 1 < i < < i, and B = j 1,..., j }, j 1 < j < < j. We write B 1 B iff i s j s for all 1 s, that is, B 1 can be obtained from B by left-pushing. Denote by LB the set of all sets obtained this way from B. Also set LB = B B LB for any B. DEFINITION.. B is said to be left compressed or stable iff B = LB. We also recall the well-nown exchange operation S ij, defined for any family B.

4 65 R. Ahlswede et al. DEFINITION.3. Set i} B\ j} if i B, j B, i} B\ j} B, S ij B = B otherwise and S ij B =S ij B : B B}. DEFINITION.4. Denote by LIn,, t the set of all stable systems from I n,, t. It is nown from the shifting technique [8] that Mn,, t = max B = max B. B I n,,t B LIn,,t DEFINITION.5. For any B we define the upset UB =B : B B } and for B we define UB = B B UB. Furthermore, recall the concept of generating sets [1]. DEFINITION.6. For any B a set gb i is called a generating i set of B, if UgB = B. Furthermore, GB is the set of all generating sets of BGB =, because B GB. DEFINITION.7. For B [1, n] denote the greatest element of B by s + B, and for B set s + B = max B B s+ B. DEFINITION.8. Let B be left compressed, i.e., B = LB. For any generating set gb GB consider LgB and introduce its set of minimal in the sense of settheoretical inclusion elements L gb. Also define G B =gb GB : L gb = gb}. DEFINITION.9. For B LIn,, t we set s min GB = min gb GB s+ gb. 1 Now we extend these definitions to a direct product of uniform sets H = 1 m in a natural way. To simplify notation we associate each bloc i = m w i 1,wi,...,wi n i } with [1, n i ] for i = 1,...,m. DEFINITION.10. For an A = A 1,...,A m m i=1 i with A i = A i, i = 1,...,m, we define m } LA = A = A 1, A,...,A m i : A i LA i, i = 1,...,m. We set also LA = A A LA. i=1

5 Intersection theorem for direct products 653 DEFINITION.11. We say that A H is left compressed or stable, if A = LA. In other words A is stable, if it is stable under exchange operations S ij with i < j inside each bloc. The generating sets of an A H and notions GA, LgA, L ga, G A one defines similarly. DEFINITION.1. For A = A 1,...,A m m i=1 i, A i = A i, denote the greatest element of A i in i =[1, n i ] by s i + A, and for A m i=1 i set s i + A = max A A s+ i A, i = 1,...,m. For A = A 1,...,A m, B = B 1,...,B m m i=1 i, A i = A i, B i = B i i = 1,...,m we write A B if A i B i inside the set i for all i = 1,...,m. DEFINITION.13. For A LIH we set s i min GA = min ga GA s+ i ga, i = 1,...,m. We start with simple, but important properties of generating sets. LEMMA.1. Let A I H. For any B, C ga GA there exists an 1 i m such that B C i t i. LEMMA.. Let A L ga. Then for any B m i=1 i with B A, either B L ga or there exists ab L ga such that B B. LEMMA.3. Let A H, LA = A, and ga G A. Choose E = E 1,...,E m ga such that for some 1 i m, s + i E = s + i ga, and denote by A E the set of elements of A, which are only generated by E. Then i for every A A E A [1, s i + ga] =E i ni s + ii A i : A A E } = i E. i E i LEMMA.4. Let A LIH, ga G A and let E = E 1,...,E m, F = F 1,..., F m ga have the properties 1 E i F i t i for some 1 i m, and E j F j < t j for all j = i and u E i F i,v E i F i for some u,v i with u <v. Then E i F i t i + 1. LEMMA.5. Pigeon hole principle with weight function. For B B : j B} let f : B R +. Then there exists an i [1, n], such that f B n f B. n B B i B B The proof is readily established by counting in two ways., B j B

6 654 R. Ahlswede et al. 3. THE MAIN AUXILIARY RESULTS LEMMA 3.1. Let A LH with A =MH and let n i > i t i t 1 1 r i + 1 for some i [1, m] and r i N 0}. Then 3.1 s i min GA t i + r i, if t i 3. and s i min GA 1, if t i = PROOF. We can assume that n i i t i +, 3.4 because for t i = 1 this is the condition 3.1 and in the case n i = i t i + 1t i > 1 we have from 3.1 r i i t i + 1, and hence 3. holds. We are going to prove only 3., because the proof of 3.3 is just a step-by-step repetition. The proof is more complex than its predecessor in [1]. However, being based to a large extent on the same ideas and methods, we can omit some details. W.l.o.g. we prove the lemma for i = 1. Let us have for some ga G A s + 1 ga = s 1 minga and let us assume in the opposite to 3. that s + 1 ga = l>t 1 + r We shall show that under the assumptions 3.1 and 3.5 there exists an A I H with A > A =MH, which is a contradiction. Towards this end we start with the partition ga = g 0 A g 1 A, where g 0 A =B ga : s + 1 B = l} and g 1A = ga\g 0 A. Obviously, for every B g 0 A and C g 1 A B\l} C i t i holds for some i, 1 i m see Lemma.1. As GA G A, we observe that omission of l from any E g 0 A destroys the intersection property, that is, there exists an F g 0 A, such that E\l} F i < t i for all i, 1 i m. The elements in g 0 A have an important property, which follows immediately from Lemma.4. P 1 For any E = E 1,...,E m, F = F 1,...,F m g 0 A with E 1 F 1 =t 1, and E i F i < t i for i =,...,m necessarily E 1 + F 1 =l + t 1.

7 Intersection theorem for direct products 655 Moreover, we have also the property P For any E, F g 0 A with E 1 + F 1 =l + t 1 holds for some i, 1 i m. Partition now g 0 A in the form E\l} F\l} i t i R i 0 i l g 0 A = with R i =F g 0 A : F 1 =i} and consider the set R i =F : F = F\l}; F R i }. Thus R i = R i and for any F R i F 1 =i 1. We shall prove that under conditions 3.1 and 3.5 all R i and hence R i are empty. As n 1 > 1 t 1, we notice that the equation E 1 + F 1 =t 1 + l for E = E 1,...,E m, F = F 1,...,F m g 0 A implies E 1 > 1 n 1 l, F 1 > 1 n 1 l. Suppose that R i = equivalently R i = for some i. We distinguish two cases a i = t 1+l and b i = t 1+l. Case a: We consider generating sets and f 1 = g 1 A g 0 A\R i R l+t1 i R i 3.6 f = g 1 A g 0 A\R i R l+t1 i R l+t 1 i. We now from properties P 1 and P that f 1 and f satisfy Lemma.1. Hence, we have The desired contradiction will tae the form B i = U f i H I H for i = 1,. A < max i=1, B i. 3.7 The negation of 3.7 is A B i 0 for i = 1,. 3.8 Let z resp. y be the number of those elements of A, which are generated only by R i resp. R t1 +l i, and let z resp. y be the number of those elements of B 1 resp. B, which are generated only by R i resp. R t 1 +l i. From Lemma.3 it follows that for some z 1, y 1 N, n1 l n z = z 1 and y = y 1 i 1 1 l l t 1 + i and similarly we obtain z n1 l + 1 z 1 1 l + 1 and y y 1 n 1 l l t 1 + i

8 656 R. Ahlswede et al. Actually, equalities hold, but they are not needed here. Hence 3.8 is equivalent to Using 3.9, 3.10 one obtains z + y z 0, z + y y n 1 + t 1 1 in 1 l 1 + i 1 i l t 1 + i + 1. However, this is false, because n 1 1 t 1 + and consequently n 1 +t 1 1 i > 1 i +1 as well as n 1 l 1 + i > 1 l t 1 + i + 1. Case b: i = l+t 1. Let T = and consider the partition where and the partition QT = E 1 [1,l] 1 : E 1,...,E m R t 1 +l R t 1 +l = QT, T T } E = E 1,...,E m R t 1 +l : E 1 = T, R t 1 +l = Q T, T T where } Q T = E = E 1,...,E m R t 1 +l : E 1 l} =T Let zqt be the number of elements of A, which are generated only by elements from QT. By Lemma.3 ii these numbers can be written in the form n1 l zqt = 1 t 1+l z 1 QT for some z 1 QT N. 3.1 Further, let zr t 1 +l be the number of elements of A, which are generated only by elements from R t 1 +l. Using Lemma.3 i and 3.1 we have z R t 1 +l = n1 l zqt = 1 t 1+l z 1 QT T T T T Now by Lemma.5 there exists a j [1,l 1] and a T T such that j T for all T T and QT l t 1 l 1 z R t 1 +l T T Let R =, and consider a new generating set f = By Lemma.4 we have Q T R t 1 +l T T ga\r t 1 +l R. U f H = B I H. }

9 We show now that under condition 3.1 Intersection theorem for direct products 657 B > A 3.15 holds, which will lead to the contradiction. Indeed, let zr be the number of elements of B, which are generated only by the elements from R. Equivalent to 3.15 is zr >z R t 1 +l The following relation similar to 3.13 can easily be verified. zr n1 l t 1+l z Q T T T Actually, equality holds here. Now 3.16 and hence 3.15 easily follow from 3.13, 3.14, 3.17, and condition 3.1. Inspection of the proof of Lemma 3.1 shows, that the following, slightly different statement also holds. LEMMA 3.. Let I =i [1, m] :t i } and let n i i t i t 1 1 r 1 +1 for i I and n i > i for i [1, m]\i. Then there exists an A LH with maximal cardinality A =MH, such that s i min ga t i + r i for i I and s i min ga 1 for i [1, m]\i. We recall the exchange operation S ij see Definition.3. DEFINITION 3.1. We say that B is invariant on T [1, n], if S ij B = B for all i, j T. LEMMA 3.3. Let A LIH, A =MH, be an optimal set from Lemma 3., i I = i [1, m] :t i } and let i t i t i 1 n i < i t i t i r i + 1 r i Then A is invariant on [1, t i + r i ] i. PROOF. It suffices to prove the lemma for the first bloc with t 1. We now from Lemma 3. that s 1 min GA t 1 + r 1. From the definition of generating sets we also now that A is invariant on [t 1 +r 1 +1, n 1 ]. Consider now the twin to set H H [n1 ] = n 1 1 [n ] [nm ] with intersection numbers t 1 = n t 1, t = t,...,t m = t m and a new set A =A 1,...,A m H : 1 \A 1, A,...,A m A}. m

10 658 R. Ahlswede et al. Clearly, A I H and A = A =MH = MH see the Remar in the Introduction. It is also clear that A is right-compressed in 1. The right side of condition 3.18 gives the relation n 1 > 1 t t 1 1 r for 1 = n 1 1, t 1 = n t 1 and r 1 = 1 t 1 r 1. From the left right symmetry and Lemma 3.1 we conclude that there exists a generating set ga such that for every E = E 1,...,E m ga necessarily E 1 [t 1 +r 1 +1, n 1 ]. Consequently A is invariant on [1, t 1 + r 1 ] and this means that A has the same property. LEMMA 3.4. Let A LIH, A =MH, be an optimal set from Lemma 3.3, and let ga G A. Then for every E = E 1,...,E m ga either E i =t i + r i or E i =0 for i I =i [1, m] :t 1 }, and E i 1 for i [1, m]\i. PROOF. Again it suffices to show that the statement holds for the first bloc 1. Moreover, we assume that 1 I, that is, t 1, because for t 1 = 1 the statement holds, according to Lemma 3.1. From Lemma 3. we now that, for every E = E 1,...,E m ga, necessarily E 1 [1, t 1 + r 1 ]. Moreover, it follows from the proof of case a in Lemma 3.1: if t 1 + r 1 E 1, then necessarily E 1 =t 1 + r 1. Suppose now that there exists an F = F 1,...,F m ga with F 1 =0 and F 1 = t 1 +r 1. We have t 1 + r 1 F 1. Two cases can occur: F 1 > t 1 +r 1 and 0 < F 1 < t 1 +r 1. Here we treat only the first case, because the second can be done by similar arguments. Let A F A be the set of those elements of A, which are generated only by F = F 1,...,F m ga. AsgA G A, then clearly A F =. Moreover, as 1 F 1 < n 1 t 1 r 1 this follows from F 1 > t 1 + r 1 there exists an A = A 1,...,A m A F, such that A 1 [1, t 1 + r 1 ]=F 1. Recall now the exchange operation and consider A 1 = S j,t 1 +r 1 A 1 for j F 1. According to Lemma 3.3 we have A = A 1, A,...,A m A as well. Let F = F 1,...,F m ga be an element, which generates A, that is, A UF. Clearly, t 1 + r 1 F 1, because otherwise A UF as well, and this would contradict the definition of the set A F. On the other hand, if t 1 + r 1 F 1, then necessarily F 1 = t 1 + r 1 < F 1 and this again leads to a contradiction with Lemma.. 4. FURTHER PREPARATIONS The following statement summarizes our findings in previous sections. LEMMA 4.1. Let r i, i I =j [1, m] :t j }, be integers uniquely determined in 3.18 and let us set r i = 0 for i [1, m]\i. Then there exists an A I H with A =MH, such that for any A = A 1,...,A m, B = B 1,...,B m A there is an i [1, m] for which both, hold. A i [1, t i + r i ] t i + r i and B i [1, t i + r i ] t i + r i

11 Intersection theorem for direct products 659 PROOF. Let A H be an optimal t 1,...,t m -intersecting system for which the statements of Lemma 3.3 and 3.4 hold. According to Lemma 3. the system A has a generating set ga GA, such that for each E = E 1,...,E m ga one has E i [1, t i + r i ] i i = 1,...,m. On the other hand, Lemma 3.5 says that the cardinality of E i i = 1,...,m is either t i + r i or 0. Therefore, for any E = E 1,...,E m, F = F 1,...,F m ga, to guarantee t 1,...,t m -intersection see Lemma.1, there must exist an i [1, m] such that E i = F i =t i + r i. Let now r i, i = 1,...,m, be integers defined in Lemma 4.1. For a C H we consider the following mappings: ϕ = ϕ 1,...,ϕ m : C 0, 1} m, where for C = C 1,...,C m C, C i = i C 1, if Ci [1, t ϕ i C i = i + r i ] t i + r i 0, otherwise, ϕc = ϕ 1 C 1,...,ϕ m C m and we set C =ϕc : C C}. For any C H and B 0, 1} m we define the weight wb, C: Clearly, wb, C = C C : ϕc = B}. wb, C = C. B 0,1} m It is also clear that for any B = b 1,...,b m 0, 1} m one has where wb, C m wb i, 4.1 i=1 F ri, if b i = 1 wb i = ni F ri, if b i = 0, i 4. and the F ri s are defined in 1.1. Now let C H be a set such that C I m, where I m is the set of all intersecting families in [m] to avoid an extra notation we identified [m] with 0, 1} m. Obviously C I H. Let I H, be the set of all such systems from H and denote Clearly, MH, = max C. C I H, and for any C I H, with C =MH, necessarily wb, C = MH, MH 4.3 m wb i for all B C 0, 1} m. i=1 It follows from Lemma 4.1 that the opposite to 4.3 also holds. Moreover, for any B I m with arbitrary weights h : B N satisfying hb mi=1 wb i for B B, one can find a C I H, with C = B and wb, C = hb, B B. Therefore one has the following

12 660 R. Ahlswede et al. LEMMA 4.. MH = max B I m B B wb, where for B = b 1,...,b m, wb = m i=1 wb i, and the wb i are defined in 4.. We need a special case of a result from Ahlswede and Cai [4]: Let u =u 1 u u m } be positive reals and let B I m be an intersecting family in [m]. Define ub = i B u i for B [1, m] and W B = B B ub for B [m]. We set αm, u max W B. B I m THEOREM 4.1 [4]. In a special case. Let u 1 < 1, then αm, u = W Bu 1, where Bu 1 =B [1, m] :1 B}. Finally we need the following statement, which can easily be proved. PROPOSITION 4.1. For all n i i t i 1 with n i > i t i for i = 1,...,m MH H holds. Moreover, if n i = i, t i = 1 for some i [1, m], then MH = H. 5. PROOF OF THEOREM 1.5 We say that B [m] is a star, if B =B [m] : j B} for some j [1, m]. According to Lemma 4., the proof of Theorem 1.5 can be finished by showing that the maximum in max B I m B B wb is assumed for a star. Of course, it is equivalent to show that max B m B B βb is assumed for a star, where for B = b 1,...,b m 0, 1} m and that is βb = βb i = ni m βb i i=1 i wb i, F ri F ri, if b i = 1 ni βb i = i 1, if b i = 0.

13 Intersection theorem for direct products 661 As for any n t 1, n > t Mn,, t n 5.1 holds, we conclude that βb i 1 for all i [1, m]. Moreover, since equality in 5.1 is achieved iff n =, t = 1, then, according to Proposition 4.1, we can assume that βb i <1 for all i [1, m]. Now we apply Theorem 4.1 with respect to the reals u i = βb i, i = 1,...,m, to complete the proof of Theorem 1.5. REFERENCES 1. R. Ahlswede and L. H. Khachatrian, The complete intersection theorem for systems of finite sets, Eur. J. Combin., , R. Ahlswede and L. H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A, , R. Ahlswede and L. H. Khachatrian, The diametric theorem in Hamming spaces optimal anticodes, Adv. Appl. Math., , R. Ahlswede and N. Cai, Cross-disjoint pairs of clouds in the interval lattices, The mathematics of Paul Erdös, R. Graham and J. Nesetril eds, Algorithms Combin., , Vol. I, M. Deza and P. Franl, Erdös Ko Rado Theorem years later, SIAM J. Discrete Math., , P. Erdös, Chao Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford, , P. Erdös, My joint wor with Richard Rado, in: Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., , P. Franl, The Erdös Ko Rado Theorem is true for n = ct, Colloq. Math. Soc. János Bolyai, , P. Franl, The shifting technique in extremal set theory, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., , P. Franl, An Erdös Ko Rado Theorem for direct products, Eur. J. Combin., , W. Haemers, Eigenvalue methods, in Pacing and Covering in Combinatorics, A. Schrijver ed., Tract 106, Mathematical Centre, Amsterdam, A. J. W. Hilton and E. C. Milner, Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford, , L. Lovász, The Shannon capacity of graphs, IEEE Trans. Inform. Theory, , A. Sali, Some intersection theorems, Combinatorica, 1 199, R. M. Wilson, The exact bound in the Erdös Ko Rado Theorem, Combinatorica, , Received 14 April 1997 and accepted 7 April 1998 R. AHLSWEDE et al. Faultät für Mathemati, Universität Bielefeld, Postfach , D Bielefeld, Germany hollmann@mathemati.uni-bielefeld.de