68-Erdos Ko Rado

SIAM J. ALG. DISC. METH.Vol. 4, No. 4, December 1983

1983 Society for Industrial and Applied Mathematics

0196-5212/83/0404-0001 $01.25/0

ERDIS-KO-RADO THEOREM--22 YEARS LATER*

M. DEZA" AND P. FRANKL

Abstract. In 1961 Erd6s, Ko, and Rado proved that, if a family : of k-subsets of an n-set is suchrt--lthat any 2 sets have at least elements in common, then for n large enough Irl =< (k-t). This result had

great impact on combinatorics. Here we give a survey of known and of some new generalizations andanalogues of this theorem. We consider mostly problems which were not included or were touched verybriefly in the survey papers [17], [46], [51], [61].

1. The Erd6s-Ko-Rado theorem. Let X ={x 1,". ,x,} be a finite set ofcardinality IxI n. 2x denotes the power set of X, while () stands for the set of allk-subsets of X.

THEOREM 1.1 (Erd6s, Ko, Rado [15]). Let n, k, be integers, with n > k > > 0and suppose is a family of k-subsets of X, i.e.

_(). Suppose further that any two

members of intersect in at least elements. Then, for n > no(k, l),

b) (;,--’,) ifffor some Xo () we have {F (>D" Xo c F}.Remark 1.2. Actually, in [15] the theorem was formulated for antichains with

IFI--< k, i.e. c (() kJ... L ()) and there are no F, F’ such that F F’ holds.However, this version is an easy consequence of Theorem 1.1.

We say that a family,_2x is l-intersecting if for any F, F’ -, IF t3 F’I--> holds.

We define now a special class of/-intersecting families.Let Xg be a subset of X of cardinality + 2i. Define {F (:): IF (3XI _-> + i}.

It is easy to see that for F, F’ one has IF (3 F’I--> l, i.e. is/-intersecting. Note that

max I[iff n >-(k-l+l)(l+l).O<_i<_k-I

Thus Theorem 1.1 does not hold for n < (k + 1)(/+ 1).THEOREM 1.3 [27]. Suppose >= 15, () and is l-intersecting.a) If n > (k + 1)(l + 1), then

[[ <- (-tl), and equality holds iff is of the form @o.b) If n (k -l + 1)(l + 1), then

and equality holds iff is of the form o or 1.c) There exists an absolute constant c, c < 1, such that ]:or

c (k + 1)(1 + 1) <= n < (k + 1)(l + 1)we have

(t / and equality holds iff is of the form 1.Remark 1.4. In the case 1, Erd6s, Ko, and Rado proved that I-[ -< (-) itt

n =>2k. For 2<=I =< 14, in [27] 1-I _-< (-tt) is established for n =>2(k -l + 1)(/+ 1). Theoriginal bound of Erd6s, Ko, and Rado was n -> + (k-/)(t)3. Hsieh [44] improvedthis to n => + (k -l + 1)(! + 1)(k -l). In the case 1, n 2k each maximal (i.e. non-extendable) family has maximum size (_-1).

* Received by the editors January 29, 1982, and in revised form July 15, 1982.t Centre National de laRecherche Scientifique, 15 Quai Anatole France, 75007 Paris, France.

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420 M. DEZA AND P. FRANKL

The general case would be settled by the followingConjecture 1.5 [27]. Suppose c (:) and is/-intersecting. Then

maxO<_i<_k-I

The case 2, n 2k, k even of the above conjecture was already conjecturedby Erd6s, Ko, and Rado.

The original proof of Theorem 1.1 and that of Theorem 1.3 use the so-calledexchange operation.

DEFINITION 1.6. Let - be a family of subsets of the ordered set X {x 1," ", xn},1 <_-i </" _-< n. The exchange operation Ei.j is defined by

E..(F) Fif xC:F, xF, ({x}UF-{x}):,otherwise,

Eid(;) {E,d(F): F 9;}.This operation is called also compression or pushing. The main importance of

this operation lies in the followi.g easy lemma.PROPOSITION 1.7. If 2 is l-intersecting, then Ei,I(.) is l-intersecting as well.Iterating the exchange operation Ei.i for every pair 1 <-i <j -< n, at last we obtain

a family * which is stable i.e. Eid(;*) ’* for every 1 -< <j <-n.For the case 1 of Theorem 1.1, Katona [45] gave a nice, simple proof,

Daykin [6] showed that this case is a consequence of the Kruskal-Katona theorem(cf. [52], [48], [5]).

Remark 1.8. Let us define the graph G(n, k, l) whose vertex set is () and inwhich (F, F’) is an edge iff IF fqF’[ < 1. Then Theorem 1.1 gives that for n > no(k, l)the independence number a(G(n, k, l)) (,--l). Using the linear programming boundfor association schemes of Delsarte [7], in [59] Schrijver strengthened the Erd6s-Ko-Rado theorem by proving that the capacity (cf. Shannon [60]) of G(n, k, l) is equalto (-t) for n > n’(k, l). Brouwer and Frankl [3] showed that it holds for n > k:/2. Itwould be interesting to know whether the same holds already for n => (k -l + 1)(/+ 1).

There is an exciting conjecture of Chvfital that is connected with the case 1.To state it we need a definition.

A family r is said to be a simplicial complex (ideal, hereditary system, down-set)if G cF " implies G .

For x eX let (x) denote {F e ’: x e F}.Conjecture 1.9 (Chvfital [4]). Suppose r is a simplicial complex, 2x,

and any two elements of have nonempty intersection. Then I 1-<-maxx x ](x)l.Remark 1.10. Chvital [4] proved that the conjecture is true for stable families

(with respect to the exchange operation). He also showed that the correspondingstatement fails for > 1. Kleitman [51] proposes an interesting new approach to thisconjecture.

2. Close relatives: stability of the extremal families, degree conditions, nonuni-form case. One of the most natural questions to ask with respect to Theorem 1.1 is"what happens if we exclude the optimal families, i.e. if we suppose there is no/-set

contained in all the members of the family. This problem was solved for the caseIF OF’[ _-> 1 i.e. 1 by Hilton and Milner [43] and for the general case in [25]:

THEOREM 2.1. Suppose ;c (), ; is 1-intersecting, but ItqFFl < l; moreoveris maximal with respect to these constraints. Then for n > no(k, l) (no(k, 1)= 2k):a) k > 2l + 1 or k 3, 1. There exist D1, D2 c X, D1 (3D2 , IDol-- l, [D[-

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ERDtS-KO-RADO THEOREM 421

k + 1 such that

xb) k _-< 2l + 1. There is a D (l+) such that

Another natural question was asked by Erd6s, Rothschild, and Szemerdi (cf.Erd6s [16]). Let c be an absolute constant, 0 <c < 1, and suppose along with theconditions of Theorem 1.1, that every element x eX is contained in at most c] sets,i.e. the degree ol every point is Ncll. What is the maximum of I and which arethe extremal systems as functions of c ? This question was completely answered forseveral choices of c (c e or > c e in [25], c < by Ffiredi [35])"

THEOREM 2.2. Suppose (), F F’ and for every x X, [(x )l c.Moreover, suppose is maximal subject to these constraints. Then for n > no(k, c),c > , one of the following possibilities holds"

a) < c < 1. There exists D () such that

b) c ]. There exists D e () such that

c) 53- < c < . There exists an c (:), I[ 1 O, A A’ ]:or all A, A’ s, andILJAA[ <- 6 (there are 6 nonisomorph possibilities), such that

={F ()" A,AF}.d) < c . Leto be the family among those in c) which has IXo UA]= 6 and

every point ofXo has degree 5.

= {F ()" ao,aF}.e) c . The family o and Xo is as in d),

f) < c < . There exists V (), () such that {P, P, aprojective plane o order 2 on V (i.e. IP 3, IP P 1, 1 ] 7) such that

={F ()" P,P=F}.Ffiredi [35] proved the following general theorem, too.THEOREM 2.3. With the conditions of Theorem 2.2, there exists a sequence 1 c >

c2 >"" >c >... of positive constants tending to 0 as s such that for ci c <hoMs. Moreover, g a projective plane of order i-1 exists, then

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ci i/(i2-i + 1). (Note that I :l fl(n) means thatforfixed k and c there exist constantsdo, dl such that for every n we have don <-IST] <-dn.)

If we do not make restrictions on IF[ for F , we haveTHEOREM 2.4 (Katona [47], Kleitman [49]). Suppose o 2x, IF F’[ 2 for

F, F’ , and is maximal. Thena) n + 2t.

={F=X: IFlt}.b) n + 2t + 1. There exists an x X such that

{F = X: IF (X {x})l t}.

Remark 2.5. In the case l= 1, one trivially has ][2"-1 (if Fs then(X-F) ) and, as in the case n 2k of Theorem 1.1, every nonextendable family

has I 2"-.Let us also mentionTHZOEM 2.6 (Milner [56]). Suppose 2x is an l-intersecting antichain. Then

+ + Moreover, or n + even equality holds iff=2

Remark 2.7. It is shown in [24] that for n +l 2t + 1 there is only one morepossibility for equality, namely

={F (tl)’ YF}U{F ()" Y cF}, where Y ()is fixed.

TOM 2.8 (Hilton [42]). Suppose c 2x, g N IF h VF , where0 < h N n, g N min (h, n h). Moreover, is l-intersecting. Then

gNiNh

TOM 2.9 (Green, Katona, Kleitman [36]). Suppose c 2x is an antichainwith IFI n/2 and F F’ [or F, F’ e . Then

<1F F

3. Families with prescribed eardinalifies [or pairwise intersection. In this sectionwe consider the following general problem.

LetL {l, 12, , Is} be a set of integers with 0 l <.. < l < k. A family = ()is called an (n, k, L)-system if for all F F’ we have IF F’I L. What is themaximum cardinality of an (n, k, l)-system, which are the optimal families?

THEOREM 3.1 (Deza, Erd6s, Frankl [I 0]). Suppose is an (n, k, L)-system. Thena) For n > no(k, L)

n I==sk -li

b) There exists a constant c c(k, L) such that

n-l k-l

implies (l- l)[(l- l). .[(k l) and Xoe () such that Xo F,

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ERD6S-KO-RADO THEOREM 423

Remark 3.2. Theorem 1.1 is the case L {/, + 1,. , k 1}. The case of equalityin a) corresponds to perfect matroid designs, i.e. matroids in which any two fiats ofthe same rank have the same size (for survey see Deza and Singhi [11]).

Remark 3.3. The case s 1 of the above theorem was earlier proved in [8],where for c c(k, L) the best possible bound k2_ k + 2 is established. The case s 2was settled by Deza, Erd6s and Singhi [9].

The following theorem combines results of Frankl and Wilson [33], and Frankland Rosenberg [32]:

THEOREM 3.4. Suppose is an (n,k,L)-system, r>-2, integer, such thatkli(mod r) for 1,..., s. Suppose the li’s lie in altogether different congruenceclass mod r.

a) If r is a prime, then

b) If r is a prime power, then

n

c) If t 1, then

Remark 3.5. If we choose r to be a prime with r > k, then k lg (mod r) and s,thus we obtain (7), which was proved by Ray-Chaudhuri and Wilson [58]. Thecase c) improves earlier results by Babai and Frankl [1] and Deza and Rosenberg [14].

Remark 3.6. Theorem 3.4 has important applications. Here we sketch one ofthem. Let Gn be the graph whose vertices are the points in En, the Euclidean spaceof dimension n, and the edges those pairs of points whose Euclidean distance is 1.Suppose this graph has chromatic number m, and let E"= B1 t.J B2 I,.J... I,.J B, be acorresponding coloration. Let p be a prime (we shall fix it later) and let X denote theset of points x (xl,... ,x,) in E" which satisfy xi =0 for n -2p+ 1 and xi 1/x/pfor 2p-1 values of i. For x X define F(x)={i" xi 1/x/p}. Then F(x) is a (2p-1)-subset of {1,..., n}, and x,x’ are at distance 1 if and only if [F(x)fqF(x’)l=p 1holds. Applying Theorem 3.4 with k 2p 1, r p, L {0, 1,. ., p 2, p,. ., 2p 2}we obtain [X f’lBil<-(p"_) for i-1,..., m. Thus m _->(2pL1)/(p) holds. Choosingp---(2-x/-)n/4 we infer m_->(1.2)" i.e. the chromatic number of Gn is growingexponentially.

Both Theorem 3.1 and Theorem 3.4 deal with general L; they can be improvedfor particular choices of k and L. Let us denote by re(n, k, L) max I-[, is an(n, k, L)-system.

In [30] and [19] the correct order of magnitude of re(n, k, L) (i.e. upper and lowerbounds which are only a constant factor apart) is determined for k -< 7 and k 8, 9 andL is arbitrary.

In [19] the case ILl- 3 is considered. Necessary and sufficient conditions are givenfor m (n, k, L) O(n) and m (n, k, L) _-> O(n 2).

As a curiosity let us mention that re(n, 12543, {0, 112, 1233})- O(n) itt there isno projective plane of order 10.

A conjecture of Erd6s and S6s (cf. Erd6s [16]) was proved in [26]:

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THEOREM 3.7. For n >--no(k), k _-> 4,

m(n,k,{O, 2,3,..., k- 1}) (-)and ;attains this bound ifffor some x, y X we have " {F (:): {x, y } c F}.

Erd6s and Frankl have the following general conjecture.Conjecture 3.8. Suppose n > no(k, l), k > >- O.a) If k > 2l + 1, then

m(n,k,{0,1,...,k}-{/})=k-l-

and for -c (:) equality holds if and only if for some Y (l+X) we have -={F (): Y

_F}.

b) If k _<-21 + 1, then

m(n,k,{O, 1 ..., k}-{l})<(7)(2k-l-1)/(2k-l-1)= k

and equality holds for r c (7) iff them exists an (l, 2k 1, n)-Steiner system, ’, and-{F s (c): there exists an S " with F_

S}.(A (t, s, n)-Steiner system is a family of s-subsets of an n-set, such that each

t-subset is contained in exactly one member of the family.)Using Theorem 3.4b, it was proved in [33] that the inequality of Conjecture 3.8b

is true if k =< 2l + 1, and k -l is a prime power.Combining a result of [29] and part b) of Theorem 3.4, we haveTHEOREM 3.9. If k >- 3l + 2 or if k > 21 + 1 and k is a prime power, then

((n-l-ll))m(n,k,{O, 1,...,k-1}-{/})= 1+o(1)k-l-

The determination of the correct order of magnitude of re(n, k, L) seems to behopeless. We cannot even decide whether there exists some integer k which is not ofthe form 2 1 or 3b but m (n, k, {0, 1, 3}) > cn3for some positive constant c (any suchk should satisfy k -= 1 or 3 (mod 6)).

4. More generalizations of the Erd6s-Ko-Rado theorem for systems o| finite sets.THEOREM 4.1 (Erd6s [18]). Let s >=2 be an integer, -= (c), and suppose ;does

not contain F1, ", Fs such that Fi f’)F. f for all <- <f <- k. Then for n > no(k, s)

].1<()_(n -s+l)k

and equality holds ifffor some Y (sx_). r= {F ()" F fq Y }.Remark 4.2. The case s 2 corresponds to the case 1 of Theorem 1.1. As to

the bound no(k,s) we conjecture no(k,r)<ckr but only no(k,s)<-2k3s (Bollobas,Daykin, Erd6s [2]) and no(k,s)<-c’ks 2 ([19]) are known. In [31] under much moregeneral conditions a weaker estimate [[ < ken k- is proved.

Remark 4.3. For the case > 1 of Theorem 1.1., Hajnal and Rothschild [41] gavethe corresponding generalization. In Deza, Erd6s, Frankl [10] asymptotic bounds, wereobtained for the more general case: if among any s members F, ., F of there aretwo with [F fq F.[ L.

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ERDOS-KO-RADO THEOREM 425

Kleitman [50] considered the problem of maximizing Il such that ’c 2x, "contains no s pairwise disjoint sets, i.e. the nonuniform case. He obtained best possiblebounds for n -= 0,-1 (mod s).

There are several generalizations of Theorem 1.1 to multiple intersections. Welist some of them below.

THEOREM 4.4 [23]. Suppose (), any members of have nonemptyintersection and n >-(t/(t- 1))k. Then

Remark 4.5. The restriction n >= (t/ (t -1))k is best possible, since for k >((t-1)/t)n any sets of cardinality k have nonempty intersection. It would be veryinteresting to obtain best possible bounds also for the case > 1, analogously withTheorem 1.3. The case " is an antichain was solved for n > 1.000, 3 in [23] and for-> 4 by Gronau [38], who also settled most of the remaining cases for 3 (cf. Gronau

[39], [40]).XLet X (+a) and define , {F ()" IF f’l XI-> + (t- 1)i}.

Confecture 4.6. Suppose "c (:), /F1, ", Ft , IF fq’ Cl F,I --> holds. Then

<-- maxOik

This conjecture would generalize Conjecture 1.. With the above notation, define{A X: [AfqX[>-_l+(t-1)i}. Then2x and for every A1,... ,A, of

course, [A (3.. f3 At[ _-> holds.Conjecture 4.7 [28]. Suppose 4 c 2x and A1, ,A , [A (q" "Atl->-

holds. Then

lai <-- maxO<--<_i_(n-l)/t

In [28], this conjecture is proved for _<-t2t/150. Theorem 2.4 shows that it holdsfor 2. In [20], it is proved for 2, 3(lag[ <_-I,1 2-=).

Most of the theorems could have been formulated for unions instead of intersec-tions-it suffices to take the complement of the sets. However, if we make restrictionson both unions and intersections at the same time, interesting new problems arise.The following result was conjectured by Katona in [46].

THEOREM 4.8 [22]. Let n > >= 1 and suppose c 2x such that tF1, F2 ,F f"l F2 # , IF1 tAF2l<-n -l.

a) If n 1 + 2t, then

b) If n l + 2t + l, then

Here we give a new and short proof of Theorem 4.8 using a result of Chvital(mentioned in Remark 1.10) and Theorem 2.4 (Katona, Kleitman).

Let be as in the theorem. Define , {G" :IF , G c_c_ F}. Then c. and, is an ideal, the ideal generated by r. Moreover for F, F’ e ,, IF CI F’I-< n holds.

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Let us apply repeatedly the exchange operation (cf. Definition 1.6) to ,. Atlast we obtain a family which is stable under this operation. Let it be M, and let

’ M, the family corresponding to . Of course I1 Il, satisfies the assumptionsof the theorem, and for all A, A’ .., IA LI A’] -< n holds.

As . is hereditary, stable, and any two elements of ’ have nonempty intersec-tion, we may apply the above cited result of Chvfital: there exists x e X, such thatI,,(x) {,A e,," x

Set {A-{x}" A e,,(x)},For A, A’ e ,, IA [-J A’] _-< n 1 I. Thus for B, B’ e , IB fq B’I => holds. We may

apply Theorem 2.4 to

For the general case we haveConjecture 4.9 [21]. Let l, r >-2 be integers. Suppose 2x satisfies IF LIF’I -<

n l, IF f’l F’I => r for all F, F’ e . Assume, moreover, that ]1 is maximal. Then thereexist integers s, => 0 and disjoint sets A, B X2 such that ]A] 2s + 1, ]BI 2t + r and

--{F 2x. IFAl<-s, lFf3nl>-t+r).Let us note that Winkler has recently stated the same conjecture (cf. [61]).

5. Algebraic generalizations. The subsets ofX {x 1, , x,} can be representedby 0-1 sequences, their characteristic functions:

Let A be a subset of X, and define [: {1, 2, n}-> {0, 1} by [A(i) 1 iff XiThen ]A fq A’] is the number of nonzero positions in which the two sequences agree.For fixed integers k, s, >- 1, it is natural to ask what is the maximum number of

functions f: {1, 2,. ., n}-> {0, 1, .., s} such that for any two functions [, f’(*) I{i:f(i) 0}1- k,

(**) I{i: f(i) f’(i) 0}[ >_- 1.We shall denote by Tn.s the set of all functions f: {1,..., n}->{0,..., s}.The ErdSs-Ko-Rado theorem gives the answer for s 1, n >-no(k, l).Let us define the pushing-up operation in the ith position Pi for a family - of

functions by

e(f)(/)=f(/) for/" i,

s if f(i) 0 and the function defined by setting f(i)- s is not in ,Pi(f)(i)

f(i) otherwise.

Saying it with words, we replace f in f(i) 0 by a function which differs from itonly in the ith position, where its value is s, if this new function was not yet in the system.

We set, of course, Pi (,) {Pi (f)"It is easy to check that the number of nonzero positions of f is the same as that

of P(f). If satisfies (,) and (**), then so does P(’). Repeated application of theseoperations, for 1 -< _<-n yields a family which is stable under the application of Pii.e. Pi(’) " for 1 =<i --< n.

For f " define B (f) {i" f(i) s},PROPOSITION 5.1. If satisfies (.) and (**), thena) IB (f)l <= k for f e :r,b) [B (f) n B (f’)l >- for f, f’Proof. Let us define the function g by g(i) =f(i) if f(i) 0 or f’(i) s and g(i) s

otherwise. As is stable, g . The two functions g and f’ agree in a nonzero position] if g(f)=f’(f)=s, thus f(])=s, which means ]e(B(f)fqB(f’)), proving b), a) istrivial.

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ERDS-KO-RADO THEOREM 427

Let h be any function in Tn,s such that h has k nonzero values and B(h)= B(f)for some f. Then, in view of Proposition 5.1 h has at least nonzero commonvalues with every f’ . Thus if was maximal then h . This means that canbe defined via {B (f)’ f o} in the following way

T={f e T,,s" B (f)el}.

On the other hand, let Yd be a subset of 2{x’’"}, and suppose Y3 is/-intersectingwith IBl<-k for B eYd. Let (Ya)= {fe T,,," B(f)e ; I{i’f(i)#O}l=k}.

Then () satisfies (.) and (**). Moreover set br I{B" B @, IBI r}l. Then itfollows that

(***)I()l- . b (s- 1)-O<__r<=k k

THEOREM 5.2. If 1 and ;satisfies (,) and (**), then

s -a ---I{f r,,s. I{i" f(i) # 0}l k, f(1) s}l.

Proof. In view of the above preparations, we may assume =() for ac2{x’2,’"} with [B[<-k and IB B’I_-> 1 for all B,B’e. If k >n/2 and/" is some

integer with n/2 <f <=k. Then bj+b,_j<=(’}) (since from any set and its complementat most 1 is in ). Also bn-. <= (n"--]-x) (from Theorem 1.1). Together this yields:

() (n --(n --J))(s--1)k-(-i)b. n-/’k_/. (s- 1)k-+b"-’k (n-f)

_<( -)()nn-j (s_l)k_+( n-1 )(n-(n-j))(s_l),__,,n-1Using these inequalities and br -<_ (-1) for r <-n/2, we obtain

o<=<= j 1 k-i ks

Remark 5.3. This theorem was first stated by Meyer [54]. However his proofwas incomplete. Hence, we included this proof, which was given by the second authorin 1976, but was never published. One can also prove the uniqueness of the extremalsystems unless s 1, k n. The case > 1 is more complicated. The above proof yields

THEOREM 5.4. Suppose n > no(k, I), and satisfies (*) and (**). Then,(,-_l)s -l, and equality holds ifffor pairs ofintegers (i, ), 1 <-ix <" < il <- n, 1 <- f, <- s,1 <= <- we have

; {f T,,s: ]:(it) for 1 <-_ <= l, f satisfies (.)}.

Frankl and Fiiredi [34] provedTHEOREM 5.5. Suppose k n, satisfies (.) and (**), and has maximal

cardinality. Then for >-_ 15

Il-s-’ ill l/<=s.

Remark 5.6. The proof gives an interesting application of Theorem 1.3. As amatter of fact Theorem 1.3 implies via (***) that I l/s _-<(1+o(1))s-’ from which

<- s is deduced quite easily for every n, i.e. we deduce an exact result from anasymptotic one. Recently Moon [59] gave a nice proof for Theorem 5.5. Howeverher proof works only in the case s => + 2.

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Remark 5.7. It is relatively easy to determine the maximal number of(1, 2,..., s) sequences ot length n such that they agree in at least one out of any rconsecutive positions. This maximum is attained by the sequences which have 1 inthe r’th, 2rth, , [n/r] rth positions.

An analogous problem ot coding type was considered by Miczo [55].A special type of functions are permutations. Let us denote by R(n, >=l) the

maximum number of permutations of {1, 2,..., n} such that any two permutationsr(i), p(i) agree in at least positions (i.e. 7rp

-1 has at least fixed-points). Let S,denote the symmetric group on n elements. For zr E $, define F(zr) {i E {1, ., n}:zr(i) 1}. Define further P(k, s) {zr S,: IF(zr) tq {1,. ., k}l--> s}. We use the notation7rP {p: p e e}.

THEOREM 5.8 [13]. Suppose n > no(n -1) and P is a set ofpermutations such thatany two members ofP agree in at least positions. Assume moreover, that IPI is maximal.

a) If n + 2t. Then there exists a 7r S, such that

P 7rP(n, t).

b) If n + 2t + 1. Then them exists a z: e S, such that

P 7rP(n 1, t).

For the case n > n0(l), we made in [13] the following conjecture.Conjecture 5.9. If n > no(l), then

R (n, >_-l) (n -/)!

Remark 5.10. In [13] we proved that the above conjecture is true if in $, thereexists a sharply/-transitive set Q of permutations, i.e. for any two ordered/-subsetsof $, there is exactly one permutation in Q, mapping the first on the second. Inparticular for 1, n arbitrary; 2, n a prime power; 3, n a prime power plusone. At the time being we can only prove the following result.

THEOREM 5.11. Suppose P S, and any three elements ofPhave at least positionsin common. Then for n >- no(l) we have [P[ _-< (n l)!

Of course, this theorem yields the same bound if we replace 3 by t, >_-3.

The determination of R (n, -_>l) would be settled by the following conjecture.Conjecture 5.12 [12]. There exists a family of subsets of {1, 2,..., n} such

that IF f’)F’[ _>- for F, F’ r and

R (n, =>l) ]P {r E S.: F(rr) E -}1.

Taking into consideration that every permutation is a function 7r: (1, 2," , n){1,..., n} Theorem 5.5 yields, for l=> 15, R(n, >-l)<-n "-l--- (n-l)! e"-.x/Trn.

S, can be made to a metric space by defining d(Tr, p)= n-IF(Trp-)l. With thisterminology we are concerned with anticodes i.e. sets with given maximal distance d.Theorem 5.8 states that for n > no(d) any maximal sized anticode is a sphere or nearsphere. Note the analogy between Theorem 5.8 and Theorem 2.4.

Let us denote by R (n, L) the maximum cardinality of a set of permutations P___S,

such that for every r#pEP we have IF(zrp-1)[EL (L ={/1,""",/s}_{0,""", n-2}).Kyota [49] gave a very elegant argument showing that if P is a group, then IPI--<IIi=x (n -li). For L {0, 1, ., l- 1} equality corresponds to sharply/-transitive per-mutation groups.

Intersection problems can be considered for vector spaces as well. Hsieh [44]proved the following

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THEOREM 5.13. Let be a family of k-dimensional subspaces of V(n, q), ann-dimensional vector space over the finite field of q elements. Suppose for all A, B ;we have dim (A f’)B)>_-/>0. Assume k <-_(n -1)/2, or k <(n -1)/2 in the case q =2,> 1. Then

Il < tin-’,]=thelq number of (k l)-dimensional subspaces of V(n l, q).tk taq

Greene and Kleitman [37] showed, employing a method of Katona [45], thatTheorem 5.13 remains true for !- 1 and n 2k. Hsieh’s proof is long and involvesa lot of calculation. Here we sketch how the case 1 can be deduced quickly usingthe special case n 2k.

We apply induction on n, starting from 2k. Let n be the smallest value for whichthe statement is not proved yet. Let vl,/32,’’’, On form a basis for V(n,q). SetV (vl,..., vi), the subspace generated by va,..., vi. For u V, v e V,- V/, wedefine an exchange operation.

Let A e ’, such that u A, v e A, then choose an arbitrary k-subspace of (A, v)containing u but not v, and which is not yet in " (if such a subspace exists), replaceA by this subspace (simultaneously for all such A ’). We ke.ep on a.pplying thisoperation for all possible pairs (u, v) until we obtain a stable set " (i.e. " is invariantunder the exchange operation). We claim that has the property A fib {0}, andeven that any A,B have nontrivial intersection in V,_a. In fact, suppose thecontrary, and let Ao A f’) V,_x, Bo B f’) Vn-x, 0 # w A fq B. As dim Ao+ dim Bo2k-2 <n- 1 there exists a k-dimensional subspace B’ of V,_I such that BocB’,Aof’)B’ ={0}. Let u B’-Bo. Then the application of the exchange operation u, vcould exchange B for B’. But is stable, thus B’. Moreover, B’ f’IA ={0}, acontradiction. Now the result follows by induction.

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68-Erdos Ko Rado