The Complete Intersection Theorem for Systems of Finite Sets

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1 Europ. J. Combinatorics (1997) 18, The Complete Intersection Theorem for Systems of Finite Sets R UDOLF A HLSWEDE AND L EVON H. K HACHATRIAN 1. H ISTORIAL B ACKGROUND AND THE N EW T HEOREM We are concerned here with one of the oldest problems in combinatorial extremal theory. It is readily described after we have made a few conventions. denotes the set of positive integers and, for i, j, i j, the set i, i 1,..., j is abbreviated as [ i, j ]. For k, n, k n, we set 2 [ n ] F : F [1, n ], [ n ] k F 2 [ n ] : F k. (1. 1) A system of sets ( [ n ] k ) is called t - intersecting if and I ( n, k, t ) denotes the set of all such systems. The investigation of the function A 1 A 2 t for all A 1, A 2, (1. 2) M ( n, k, t ) max, 1 t k n, (1. 3) I ( n,k,t) and the structure of maximal systems was initiated by Erdö s, Ko and Rado. According to reference [7], they had already proved the following theorem by the year 1938, although it was only published in 1961 in their famous paper [9]. T HEOREM EKR. For 1 t k and n n o ( k, t ) ( suitable ) M ( n, k, t ) n t. (1. 4) k t Clearly, the system ( n, k, t ) A [ n ] k : [1, t ] A (1. 5) has cardinality ( n t k t ) and is therefore optimal for n n o ( k, t ). The smallest n o ( k, t ) for which this is the case has been determined by Frankl [10] for t 15 and subsequently by Wilson [13] for all t : n o ( k, t ) ( k t 1)( t 1). (1. 6) Moreover, for n ( k t 1)( t 1), there is up to obvious permutations on the ground set [1, n ] only one optimal system. In the present paper we settle all of the remaining cases n ( k t 1)( t 1) / 97 / $ / 0 ej Academic Press Limited

2 126 R. Ahlswede and L. H. Khachatrian In particular, we prove the so-called 4 m - Conjecture (Erdö s, Ko and Rado, 1938 ; see [7, page 56] and the survey [6]) Thus, obviously, M (4 m, 2 m, 2) F [4 m ] 2 m : F [1, 2 m ] m 1. (1. 7) M (4 m, 2 m, 2) m 2 m 2. (1. 8) 2 m m The previously best upper bound on M (4 m, 2 m, 2) is due to Calderbank and Frankl [5]. There is a natural extension of the 4 m -Conjecture in terms of the systems to all possible parameters. i F [ n ] k : F [1, t 2 i ] t i for 0 i n t 2 G ENERAL C ONJECTURE [10]. For 1 t k n, (1. 9) M ( n, k, t ) max 0 i n t 2 i. (1. 10) Notice that for n 4 m, k 2 m, t 2 the maximum is assumed for i m 1 and so the 4 m -Conjecture is covered. For n ( k t 1)( t 1) the maximum is assumed for i 0. A further step towards proving the General Conjecture was taken in [10] for t 15, where the cases 0 8( k t 1)( t 1) n ( k t 1)( t 1) are settled. Here 1 is (up to permutations) the only optimal system. Some other cases have been settled in [12]. Our main result establishes the validity of the General Conjecture and provides an even more specific answer concerning uniqueness. T HEOREM. For 1 t k n : (i) With ( k t 1)(2 r t 1 1 ) n ( k t 1)(2 t 1 r ) for some r 0, we ha e M ( n, k, t ) r and r is up to permutations the unique optimum. ( By con ention, r 0.) (ii) With ( k t 1)(2 r t 1 1 ) n for r 0 we ha e M ( n, k, t ) r r 1 and an optimal system equals up to permutations either r or r 1. t 1 r for C ONVENTION. In the sequel, we write 1 2 if the systems 1 and 2 are equal up to permutations. R EMARKS. (1) It suf fices to treat the cases because for n 2 k t the whole system ( [ n ] k ) is t -intersecting. n 2 k t, (1. 11)

3 The complete intersection theorem 127 (2) Our method of proof follows ideas in our work in Number Theory (see [2], [3], [4]). They led to the concept of generating sets in Section 2 and related pushing techniques in later sections, which go considerably beyond known techniques in Combinatorial Extremal Theory (see [11]). (3) The t -intersecting systems in I ( n, k, t ) can be understood as systems with a diameter less than 2 k 2 t in the Hamming distance. Instead of Intersection Theorems one can then speak of Diametric Theorems a concept of geometrical meaning. This and relations to Isoperimetric Theorems have been discussed in [1]. (4) In [8], Erdö s mentions the 4 m -Conjecture as the last open problem from [9]. We begin with well-known notions. 2. G ENERATING S ETS AND T HEIR P ROPERTIES D EFINITION For A 1 i 1, i 2,..., i s ( [ n s ] ), i 1 i 2 i s, and A 2 j 1, j 2,..., j s ( [ n s ] ), j 1 j 2 j s, we write A 1 A 2 if f i l j l for all 1 l s ; that is, A 1 can be obtained from A 2 by left - pushing. Furthermore, let ( A 2 ) be the set of all sets obtained this way from A 2. Also, set ( )! A ( A ). D EFINITION [ n ] is said to be left compressed if f ( ). D EFINITION We denote by LI ( n, k, t ) I ( n, k, t ) the set of all left compressed systems from I ( n, k, t ). It is well-known and easily follows with the shifting technique of [9] that M ( n, k, t ) max max. (2. 1) I ( n, k, t ) LI ( n, k, t ) D EFINITION For any B 2 [ n ] we define the upset ( B ) B 2 [ n ] : B B. More generally, for 2 [ n ] we define the upset Now we introduce new concepts. ( )! ( B ). B D EFINITION For any ( [ n k ] ) a set g ( )! i k ( [ n i ] ) is called a generating set of, if ( g ( )) ( [ n k ] ). Furthermore, G ( ) is the set of all generating sets of. ( G ( ), because G ( )). A first auxiliary result is readily established. L EMMA 1. For I ( n, k, t ), n 2 k t, E 1 E 2 t for E 1, E 2 g ( ) G ( ).

4 128 R. Ahlswede and L. H. Khachatrian P ROOF. Since 2 k t n, E 1 E 2 t would allow k -element extensions of E 1 and E 2 which are not t -intersecting. Next, we introduce further basic concepts. D EFINITION For B b 1, b 2,..., b B [1, n ], b 1 b 2 b B, write the largest element b B as s ( B ). Also, for 2 [ n ] set s ( ) max s ( B ). B D EFINITION Let ( [ n ] k ) be left compressed, i. e. ( ). For any generating set g ( ) G ( ), consider ( g ( )) and introduce its set of minimal (in the sense of set-theoretical inclusion) elements * ( g ( )). Furthermore, define G * ( ) g ( ) G ( ) : * ( g ( )) g ( ). (Notice that G * ( )). We continue with simple properties. L EMMA 2. For a left compressed ( [ n ] k ) and any g ( ) G ( ) : (i) * ( g ( )) G ( ) ; (ii) s ( * ( g ( )) s ( g ( ))) ; (iii) for A * ( g ( )) and B A, we ha e either B * ( g ( )) or there exists a B * ( g ( )) with B B. The next important properties immediately follow from the definition of G * ( ) and the left-compressedness of. L EMMA 3. For a left compressed ( [ n ] k ) and g ( ) G * ( ), is a disjoint union where! ( E ), E g ( ) ( E ) B [ n ] k : B E B 1, B 1 [ s ( E ) 1, n ], B 1 k E. L EMMA 4. For a left compressed ( [ n ] k ) and g ( ) G * ( ), choose E g ( ) such that s ( E ) s ( g ( )) and consider the set of elements of which are only generated by E ; that is, E ( ( E ) ( g ( ) E )) [ n ] k. Then e ( E ) and E n s ( E ) k E. L EMMA 5. Let LI ( n, k, t ), g ( ) G ( ), and let E * 1, E 2 g ( ) ha e the properties i E 1 E 2, j E 1 E 2

5 The complete intersection theorem 129 for some i, j [1, n ] with j j. Then E 1 E 2 t 1. Finally, we use the following convention. D EFINITION For LI ( n, k, t ), we set s m i n ( G ( )) min s ( g ( )). g ( A ) G ( ) 3. T HE M AIN T WO A UXILIARY R ESULTS L EMMA 6. Let n 2 k t, LI ( n, k, t ), with A M ( n, k, t ), and let n for some r 0. Then ( k t 1)( t 2 r 1) ( k t 1) 2 t 1 (3. 1) r 1 r 1 s m i n ( G ( )) t 2 r. (3. 2) P ROOF. We can assume that n 2 k t 2, because in the case n 2 k t 1 we have, from (3. 1), r k t 1, and hence (3. 2) trivially holds. By Lemma 2 we have, for some g ( ) G * ( ), s ( g ( )) s min ( G ( )). Now assume that, in the opposite to (3. 2), s ( g ( )) t 2 r for some 0. (3. 3) We shall show that under the assumptions (3. 1) and (3. 3) there exists an A I ( n, k, t ) with M ( n, k, t ), which is a contradiction. For this, we start with the partition g ( ) g 0 ( ) g 1 ( ), where g 0 ( ) B g ( ) : s ( B ) t 2 r and g 1 ( ) g ( ) g 0 ( ). Obviously, for every B 1 g 0 ( ) and B 2 g 1 ( ) we have ( B 1 t 2 r ) B 2 t. (3. 4) The elements in g 0 ( ) have an important property, which follows immediately from Lemma 5 : (P) For any E 1, E 2 g 0 ( ) with E 1 E 2 t, necessarily E 1 E 2 2 t 2 r. Now, we partition g 0 ( ) according to the cardinalities of its members : g 0 ( )! i, i g 0 ( ) [ n ]. t i t 2 r i (Here we have used that in case t, by Lemma 1 and left-compressedness, necessarily t [1, t ] g ( ) and (3. 2) holds. Furthermore, if t 2 r, then t 2 r [1, t 2 r ] g ( ), and by optimality of necessarily r 0 and again (3. 2) holds. ) Of course, some of the i s can be empty.

6 130 R. Ahlswede and L. H. Khachatrian Next, we omit the element t 2 r ; that is, we consider i E [1, t 2 r 1] : E t 2 r i. So i i and, for E i, E i 1 ( i ). From property (P) we know that, for any E 1 i, E 2 j with i j 2 t 2 r, E 1 E 2 t. (3. 5) We shall prove that (under conditions (3. 1) and (3. 3)) all i s are empty. First we notice that the equation E 1 E 2 2 t 2 r, for E 1, E 2 g 0 ( ), implies that E j k ( ) for j 1, 2, because otherwise, for instance, E 1 k ( ) implies that E 2 2 t 2 r E 1 n k t, which, together with E 2 k, contradicts n 2 k t. Hence, if for all i, i k ( ), we have i, then ( g ( ) g 0 ( )) ( t i t 2 r i ) I ( n, t ), the set of all (unrestricted) t -intersecting systems in 2 [ n ], ( ) ( [ n k ] ), and s ( ) s ( g ( )), which is a contradiction. Suppose, then, that for some i, k ( ) i t 2 r, i or, equivalently, i. We distinguish two cases : (a) i (2 t 2 r ) / 2 and (b) i (2 t 2 r ) / 2. Case ( a ). We consider the sets f 1 g 1 ( ) ( g 0 ( ) ( i 2 t 2 r i )) i, f 2 g 1 ( ) ( g 0 ( ) ( i 2 t 2 r i )) 2 t 2 r i. We know already (see property (P) and (3. 5)) that and hence f 1, f 2 I ( n, t ) i ( f i ) [ n ] k I ( n, k, t ) for i 1, 2. The desired contradiction will take the form max i. (3. 6) i 1, 2 We consider the set 1. From the construction of f 1 and i, it follows that 1 consists of those elements of ( [ n k ] ) which are extensions only of the elements from 2 t 2 r i. We determine their number : 1 ( ( 2 t 2 r i ) ( g ( ) 2 t 2 r i ))) [ n ] k and, by Lemma 4, 1 2 t 2 r i. (3. 7) k 2 t 2 r i Symmetrically, we consider the set 1. Let B 1 be any element of i, so B 1 i 1. Clearly, B 1 g ( ), because B 1 t 2 r g ( ) and g ( ) G * ( ). Therefore, for every A ( [ n ] k ) in the form A B 1 B 2, where B 2 k i 1 and

7 The complete intersection theorem 131 B 2 [ t 2 r 1, n ], we have A 1 and A, because s ( g ( )) t 2 r. We also notice that, for B 1, B 1 i, B 1 B 1, we have Therefore Analogously, we have B 1 B 2 B 1 B 2 for all B 2, B 2 [ t 2 r 1, n ]. 1 i i. (3. 8) k i 1 k i 1 2 i, (3. 9) k i 2 2 t 2 r i k 2 t 2 r i 1. (3. 10) Actually, it is easy to show that there are equalities in (3. 8) and in (3. 10). However, that is not needed here. Now (3. 7) (3. 10) enable us to state the negation of (3. 6) in the form i 2 t 2 r i k i 1 k 2 t 2 r i and 2 t 2 r i k 2 t 2 r i 1 i. k i (3. 11) Since, by assumption, i, we can also assume that 2 t 2 r i, because otherwise the first inequality in (3. 11) is false. Furthermore, (3. 11) implies that ( n t k i )( k i ) ( k i 1)( k 2 t 2 r i 1). However, this is false, because n 2 k t 2 and, consequently, n t k i k i 1 as well as k i k 2 t 2 r i 1. Hence, (3. 6) holds, in contradiction to the optimality of. Therefore, necessarily, i for all i (2 t 2 r ) / 2. (In this case (a) we did not use the condition (3. 1), and not even the condition 0). Case ( b ). i (2 t 2 r ) / 2. Here, necessarily, 2 3. We consider the set t r / 2 and recall that for B t r / 2 B t r / 2 1 and B [1, t 2 r 1]. By the pigeon-hole principle, there exists an i [1, t 2 r 1] and a t r / 2 such that i B for all B and r / 2 t 2 r 1 t r / 2 r / 2 t 2 r 1 t r /2. (3. 12) By Lemma 5, we have B 1 B 2 t for all B 1, B 2, and since by case (a) i for i t r / 2, we obtain f ( g ( ) t r /2 ) I ( n, t ). We show now that under condition (3. 1) we have ( f ) [ n ] k. (3. 13) Indeed, let us write ( g ( )) ( [ n k ] ) 1 2, where 1 ( g ( ) t r /2) ( [ n k ] ), 2 ( ( t r /2) ( g ( ) t r /2)) ( [ n k ] ) and ( f ) ( [ n k ] ) 1 3,

8 132 R. Ahlswede and L. H. Khachatrian where 3 ( ( ) ( g ( ) t r / 2 )) ( [ n k ] ). In this terminology, the following is equivalent to (3. 13) : 3 2. (3. 14) We know (see Lemma 4) that 2 t r /2 (3. 15) k t r / 2 and estimate now 3 from below. Consider any B, B t r / 2 1. Clearly, B g ( ), because B t 2 r g ( ) and g ( ) G * ( ). Hence, for every A ( [ n k ] ) in the form A B C 1, where C 1 k B and C 1 [ t 2 r, n ], we have A 3. We also notice that, for all B 1, B 2, B 1 B 2, one has B 1 C 1 B 2 C 2 for all C 1, C 2 [ t 2 r, n ]. Therefore 1 3. (3. 16) k t r / 2 1 In the light of (3. 12) and (3. 14) (3. 16), it is suf ficient for (3. 13) that r / 2 1 t 2 r 1 k t r 2 1 k t r 2 (3. 17) or, equivalently, that r ( 1) k t r 1 ( t 2 r 1), (3. 18) 2 2 which in turn is equivalent to ( k t 1)( t 2 r 1) n. r / 2 This is true by (3. 1), because is even and t 2 r 1 t 2 r 1 for 2. r 1 r / 2 Hence 0 and the lemma is proved. C OROLLARY (for the 4 m Conjecture). Let LI (4 m, 2 m, 2) and M (4 m, 2 m, 2). Then s m i n ( G ( )) 2 m. P ROOF. Just choose r m 1 and notice that (3. 1) in Lemma 6 holds : 4 m Finally, we present the second auxiliary result. (2 m 1)(2 m 1). m

9 The complete intersection theorem 133 L EMMA 7. For any ( not necessarily maximal ) I ( n, k, t ), consider any generating set g ( ) G ( ). Furthermore, for the complemented system I ( n, n k, n 2 k t ), let f ( ) be any generating set from G( ). Then, for all A g ( ) and B f ( ), we have A B n k t. (3. 19) P ROOF. Assume that (3. 19) does not hold : A B n k t 1. Choose any F A B with F n k t 1. We know that and ( B ) [ n ] ( A ) [ n ] k n k. Since n 2 k t, t 0, and n k t 1 max( k, n k ), we can choose A * ( A ) ( [ n k ] ) and B * ( B ) ( n [ n ] k ) in such a way that A * F and B * F. Consider B * and observe that A * B * ( n k t 1) ( n k ) t 1, contradicting I ( n, k, t ). 4. P ROOF OF THE 4 m -C ONJECTURE We treat this famous case here separately, even though it is covered by the proof of the Theorem. Let LI (4 m, 2 m, 2), let M (4 m, 2 m, 2) and let g ( ) G ( ) such that s ( g ( )) s m i n ( G ( )). From the corollary we know that s ( g ( )) 2 m, that is, for B g ( ) necessarily B [1, 2 m ]. We consider the complemented : A [1, 4 m ]:[1, 4 m ] A. Clearly, I (4 m, 2 m, 2), M (4 m, 2 m, 2), and is right-compressed. From the left right symmetry and the corollary we conclude that there exists a generating set f ( ) such that for every B f ( ) necessarily B [2 m 1, 4 m ]. Now, if for all B 1 g ( ) we have B 1 m 1 then, necessarily, m 1 and m 1. The same conclusion holds if, for all B 2 f ( ), we have B 2 m 1. However, if neither of these two cases occurs then, for some B 1 g ( ) and B 2 f ( ), we have B 1, B 2 m. Finally, for every A ( [4 2 m m ] ) of the form A B 1 B 2, necessarily A, a contradiction to. R EMARK 5. Instead of the last step of the proof, one can argue that B 1 B 2 2 m 2 m 2 n k t is in contradiction with Lemma 7. C ASE (i) : 5. P ROOF OF THE T HEOREM ( k t 1) 2 t 1 r 1 n ( k t 1) 2 t 1 r. (5. 1) Choose LI ( n, k, t ) with M ( n, k, t ) and let g ( ) G ( ) with s ( g ( )) s m i n ( G ( )).

10 134 R. Ahlswede and L. H. Khachatrian We know from Lemma 6 that s ( g ( )) t 2 r and we also know that, the complemented, is right compressed, an element of I ( n, n k, n 2 k t ) and satisfies M ( n, n k, n 2 k t ). An easy calculation leads from (5. 1) to n 2 k t 1 ( k t 1) 2 k t r 1 and to n 2 k t 1 n ( k t 1) 2 for k t r (5. 2) k t r ( k t 1)( n 2 k t 1) n for k t r. (5. 3) Since n 2 k t, (5. 1) implies that always k t r. Therefore, for k n k and t n 2 k t we obtain the dual relations and t 1 ( k t 1) t 1 2 n ( k t 1) 2 for r k t r r 1 r ( k t 1)( t 1) n for k t r. Using the dual version (with respect to right compressed sets) of Lemma 6, we obtain an f ( ) G( ) with the property : every B f ( ) satisfies B [ n ( t 2 r ) 1, n ] [ t 2 r 1, n ], because t n 2 k t and r k t r. Now, if for all B 1 g ( ) we have B 1 t r, then clearly r, r (for k n k, t n 2 k t ) and the optimal sets are unique up to permutations. The same sets are obtained if, for all B 2 f ( ), the inequality B 2 t r holds. If none of the cases occurs, then for some B 1* g ( ), B 2* f ( ), B 1* t r 1 and B 2* t r 1 n k r 1 and, by Lemma 7, we will have a contradiction, because B 1* B 2* n k t 2. C ASE (ii) : n ( k t 1) 2 t 1. (5. 4) r 1 Let LI ( n, k, t ) and let M ( n, k, t ). Since n ( k t 1)(2 r t 1 2 ), we conclude with Lemma 6 that s m i n ( G ( )) t 2 r 2. Now choose g ( ) G ( ) with s ( g ( )) s m i n ( G ( )). We again consider, which is again right compressed and of maximal cardinality within I ( n, n k, n 2 k t ). From (5. 4), we derive n ( k t 1) 2 t 1 r 1 t 1 ( k t 1) 2, r where k n k, t n 2 k t and r k t r. ( k t 1) 2 n 2 k t 1 k t r

11 The complete intersection theorem 135 t 1 Since n ( k t 1)(2 t 1 r ) ( k t 1)(2 r 1 ), from the dual version (with respect to right compressed sets) of Lemma 6 one has an f ( ) G( ) the elements B 1 of which satisfy B 1 [ n t 2 r 1, n ] [ t 2 r 1, n ]. Now, if for all B 2 g ( ), B 2 t r 1, then uniquely r 1 and r 1 (for k n k, t n 2 k t, r k t r ). Also, if for all B 1 f ( ), B 1 t r, then uniquely r and r. Otherwise, if for some B 2 g ( ) and B 1 f ( ), B 2 t r and B 1 t r 1, then B 1 B 2 t r t r 1 n k t 1. This again contradicts Lemma 7. This finishes the proof, if we allow only left compressed systems as competitors. We thank K. Engel for asking for an argument for not left compressed competitors. We follow an idea of [10] to prove the uniqueness, stated in the theorem. We use the well-known exchange operation S i j, i j, defined for any family 2 [ n ] as follows : for A, i ( A j ) S i j ( A ) A S i j ( ) S i j ( A ) : A. if i A, j A, i ( A j ), otherwise, P ROPOSITION. Suppose that I ( n, k, t ) and that gets transformed by finitely many exchange operations to the set r ( see (1. 9)) for some 0 r ( n t ) / 2. Then, necessarily, r, pro ided that n 2 k t 2 for t 2, n 2 k t 1 for t 2 and k t r or k t r 1, n 2 k 1 for t 1 and r 0 or r 1. ( ) P ROOF. Without loss of generality, we can assume that S i j ( ) r. (5. 5) It is clear that, if i, j [1, t 2 r ] (or i, j [1, t 2 r ]), then r and the Proposition is true. Assume, then, that i t 2 r and j n. Let 1 A : j A, i A, (( A j ) i ), 2 A : j A, i A, (( A i ) j ). Clearly, if 1, then r, and if 2, then is obtained from r by exchanging the co-ordinates i t 2 r and j n, so the proposition is true. Suppose, then, that A 1, 2, and let us show that I ( n, k, t ) (under conditions ( )). We consider [ n ] i, j H : H [1, t 2 r 1] t r 1. k 1 We observe that, for any B 1 2, B [1, t 2 r 1] t r 1 holds. This fact follows from (5. 5). Moreover, from the same assumption (5. 5) we have the following : for every H either H j 1 or H i 2. Now we form a graph G ( V, E ) as follows : V and e ( H 1, H 2 ) E if f ( H 1 H 2 ) t 1.

12 136 R. Ahlswede and L. H. Khachatrian It can be verified that the graph G ( V, E ) is connected if f the conditions ( ) hold. Hence, under conditions ( ), if 1 and 2, then there exist B 1 1 and B 2 2 with B 1 B 2 t 1, which contradicts I ( n, k, t ). P ROOF OF THE U NIQUENESS. Let n 2 k t, I ( n, k, t ) and M ( n, k, t ), and after finitely many exchange operations S i j, i j let be transformed to the left-compressed set, LI ( n, k, t ), M ( n, k, t ). We already know that r for r 0, where r is defined by the conditions in the theorem. It can be easily verified that these r s satisfy the conditions ( ) stated in the proposition, and hence that r. R EFERENCES 1. R. Ahlswede and G. Katona, Contributions to the geometry of Hamming spaces, Discr. Math., 17, (1977), R. Ahlswede and L. H. Khachatrian, Maximal sets of numbers not containing k 1 pairwise coprime integers, Acta Arithmetica LXX, 1, (1995), R. Ahlswede and L. H. Khachatrian, Sets of integers and quasi-integers with pairwise common divisor, Acta Arithmetica LXXIV, 2, (1996), R. Ahlswede and L. H. Khachatrian, Sets of integers with pairwise common divisor and a factor from a specified set of primes, Acta Arithmetica LXXV, 3, (1996), A. R. Calderbank and P. Frankl, Improved upper bounds concerning the Erdö s Ko Rado Theorem, Combin. Prob. Comput., 1, (1992), M. Deza and P. Frankl, Erdö s Ko Rado Theorem 22 years later, SIAM J. Algebra Discr. Math., 4 (4) (1983), P. Erdö s, My joint work with Richard Rado, in : Sur eys in Combinatorics, Lond. Math. Soc. Lect. Note Ser., 123 (1987), P. Erdö s, Some of my favourite unsolved problems, in : A Tribute to Paul Erdö s, A. Baker, B. Bollobá s, and A. Hajnal (eds), Cambridge University Press, Cambridge, P. Erdö s, Chao Ko and R. Rado, Intersection theorems for systems of finite sets, Q. J. Math. Oxford, 12 (1961), P. Frankl, The Erdö s Ko Rado Theorem is true for n ckt, Coll. Soc. Math. J. Bolyai, 11 (1978), P. Frankl, The shifting technique in extremal set theory, in : Sur eys in Combinatorics, Lond. Math. Soc. Lect. Note Ser., 123 (1987), P. Frankl and Z. Fü redi, Beyond the Erdö s Ko Rado Theorem, J. Combin. Theory, Ser. A, 56 (1991), R. M. Wilson, The exact bound on the Erdö s Ko Rado Theorem, Combinatorica, 4 (1984), Recei ed 2 0 June and accepted 1 4 February R UDOLF A HLSWEDE AND L EVON H. K HACHATRIAN Uni ersitä t Bielefeld, Fakultä t fü r Mathematik, Postfach , Bielefeld, Germany