Proceedings of the International Congress of Mathematicians Berkeley, California, USA, 1986

Intersection Theorems for Finite Sets and Geometric Applications

PETER FRANKL

1. Introduction. Let X be an n-element set and F C 2X a family of distinct subsets of X. Suppose that the members of F satisfy some conditions. What is the maximum (or minimum) value of |F|—this is the generic problem in extremal set theory. There have been far too many papers and results in this area to be overviewed in such a short paper.

Therefore, we will only deal with some intersection theorems. The simplest is

THEOREM 0. Suppose that F C 2X satisfies

Ff]F,^0 for allF,F'e¥. (1.1)

Then |F| < 2""1 holds.

PROOF. If F E F then (X - F) £ F implying |F| < \2n. • A family F satisfying (1.1) is called intersecting. More generally, for a positive

integer t, F is called t-intersecting if any two of its members intersect in at least t elements.

The two central theorems concerning ^-intersecting families are the Erdös-Ko-Rado theorem and the Katona theorem (cf. §§2 and 3).

The following general problem—by now rich in results and applications—was proposed by Vera T. Sós [S]:

Given a set L = {Zi,...,l3} of nonnegative integers, a family F is called an L-system if \F n F'\ e L holds for all distinct F, Ff G L.

Determine or estimate m(n,L) = max{|F|: F C 2X is an L-system}. After surveying some general results on L-systems in §4, in §5 we consider

the case of one excluded intersection size. The interest in this special case was generated by two exciting conjectures of Erdös. The first of them said that if F C ("£) avoids the intersection size I and |F| is maximal, then (for n > no(k, I)) F has all intersection sizes less than I (an /-packing) or all greater than I ((Z +1)-intersecting).

© 1987 International Congress of Mathematicians 1986

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1420 PETER FRANKL

The second said that there exists a positive e such that every F c 2 x with |F | > (2—e)n contains two sets with intersection of size [n/4\. That is, excluding [n/4j brings down the maximum size of F exponentially.

Both conjectures were settled recently. Extensions of the second conjecture led to the following geometrical result. Let A be the vertex set of a nondegenerate simplex in Rd. There exists e = e (A) > 0 such that for every partition of Rn, n>d, into less than (l+e)n classes, one of the classes contains a set A1 congruent (isometric) to A. This and similar results are discussed in §6.

2. The Erdös-Ko-Rado theorem. Let us introduce the notation (*) = {A C X: \A\ = k}.

For Ao G (*) the family A 0 = {A G (*): A0 C A} is ^-intersecting and satisfies |Ao| = (^Zj)-

THEOREM 2.1 (ERDÖS-KO-RADO [EKR]) . Letn > k > t > 1 be integers and suppose that F C (*) is t-intersecting. Then for n > no(k,t) one has

* ( : : : )

REMARK 2.2. Erdös told me that they proved this theorem already before World War II, however, at that time there was very little interest in such results, that is why it was not published until 1961.

By now, the best possible bound for no(k,t) is known:

no(fc,t) = ( f c - i + l)(t + l) .

This was proved by Erdös, Ko, and Rado for t = 1, by the author [Fl] for t > 15, and recently by Wilson [WI] for all t.

To see that (2.1) is no longer true for n < (k — t + l)(t + 1), define families Ai for 0 < i < k - t:

Let Ai G ( t £ j and A* = J4 C (£): \Afl A{\ > t + 1 | . Now A^ is J-intersecting and simple computation shows that |Ao| = |Ai |

according as n = (k — t + 1)(£ + 1) holds. The following conjecture, if true, would determine the maximum size of t-

intersecting families in general.

CONJECTURE 2.3 [Fl]. Suppose thatF C (£) is t-intersecting. Then |F| < maxo<i<fc-t |A«| holds.

~̂AtT presentTthis conjecture appears hopelessly difficult in general. However, there are some partial results (cf. [Hu, F l , and FF4]).

It is natural to ask what families are achieving equality in (2.1). For n > (k — t H-1) (t +1) , Ao is the unique family with this property (cf. [HM] for t = 1, [WI] for t > 1). In [Fl] it is proved that for t > 15 even for n = (k-t + l)(t + l) the only optimal families are Ao and Ai- This is probably true for 2 < t < 14 as well.

INTERSECTION THEOREMS FOR FINITE SETS 1421

However, the case t = 1, n = 2k is different. Then the statement of the Erdös-Ko-Rado theorem is very easy:

| p |<

Indeed, for every A G ("£), at most one of the two sets A, X — A can be present in an intersecting family. Since n = 2k, to every A G ("£) there is a unique set B G (£) with AnB = 0, namely B = X - A.

This observation makes possible the construction of very many intersecting families F C (*) with |F| = \^).

Let T and S be disjoint subsets of X, \T\ = t, \S\ = k - t + 1. Define

B1 = \Bel : (TcBandSfi5/0)or(5cruS)

Note that Bi is i-intersecting with

|Bi| =

THEOREM 2.4. Suppose that F C (*) is t-intersecting with | f| F| < t. Then for n > n\(k,t)

|F| < max{|Ai|, |Bi|} holds. (2.2)

REMARK 2.5. For t = 1, (2.2) was proved by Hilton and Milner [HM] along with n\ (k, 1) = 2k. A simple proof was given recently by Füredi and the author [FF3]. For other proofs cf. [A] and [M]. For t > 2, (2.2) was proved in [F2]. However, the value of n\(k, t) is still unknown.

An interesting class of problems was proposed by Erdös, Rothschild, and Szemerédi (cf. [El]):

Let 0 < c < 1 be a real number. Let F C (*) be an intersecting family. Let f(n,k,c) denote the maximum possible size of F if F satisfies the addi­

tional degree condition, namely that every element of X is contained in at most c|F| members of F.

Füredi [Fui] has many results on this problem. Let me cite one, which is particularly beautiful. Suppose that I > 2, Y C X, \Y\ = I2 + I + 1, and L C b+i) is the collection of all lines of a projective plane of order / on Y. Define the intersecting family F(L) by:

F(L) = | F G (*\ FnYehV

Clearly, |F(L)| = (I2 + I + l)^1!^1) and F satisfies the above condition withc = (Z + l)/(Z2 + Z + l).

THEOREM 2.6 (FÜREDI [Fui]). Suppose that c = (I + l)/(l2 + I + 1) where I > 2 is an integer, I < k. Then for n > h(k,l) one of the following two possibilities occurs.

1422 PETER FRANKL

(i) There is no projective plane of order I and f(n, k,c) = 0(nk~l~2). (ii) There exists a projective plane L of order I and f(n,k,c) = |F(L)|.

The determination of n(k, I) appears difficult. In [FF5] n(k, I) < cik is proved, where c\ is a constant depending only on I. In [FF5] many related problems are considered, e.g., for ^-intersecting families.

3. The Katona theorem. What is the maximum size of a ^-intersecting family? This problem was asked by Erdös, Ko, and Rado and it was completely solved by Katona [Kl].

Let us define K0 = {K C X: \K\ > (n + t)/2}. It is clear that for K, K1 G Ko one has \K flK'\ > \K\ + \K'\ - n > t, i.e., K0 is ̂ -intersecting. If n +1 is odd, then one can add further sets to Ko without destroying the ̂ -intersecting property. For Y an (n — l)-subset of X, define

K-KH„-f+i,/2y One can describe Ki also by Ki = {K C X: \K n Y\ > (\Y\ +1)/2}.

THEOREM 3.1 (KATONA [Kl]). Suppose that F C 2X is t-intersecting, n>t>l. Then one of the following two possibilities occurs.

(i) n + t is even and |F| < |K0|. (ii) n +1 is odd and |F| < |Ki|.

REMARK 3.2. Note that the case t = 1 is simply Theorem 0 from the intro­duction. Katona proved also that Ko and Ki are the unique optimal families for t > 2.

For the proof of Theorem 3.1 Katona proved another important theorem, concerning the shadows of ^-intersecting families. For a family F and an integer h let Afc(F) be the ft-shadow of F, i.e.,

AÄ(F) = l H G ( T ) : H C F, for some F G F j .

Let A be a (2k — £)-element set and A = (^). Then A is ^-intersecting and clearly |Afc(A)| = (2fc"*) holds for all h < k.

THEOREM 3.3 (KATONA [Kl]). Suppose that F C (£) is t-intersecting. Then for k — t < h <k one has

Note that the RHS^oT(3.1) is 1 for h = k — t or k but it is strictly greater otherwise. To illustrate the strength of this theorem, let us use it to prove the t = 1 case of the Erdös-Ko-Rado theorem. This proof is due to Katona [Kl].

Suppose that G C (^), G is intersecting, n > 2ft. We want to show that |G| < (lZ{) holds. Consider F = {X - G: G G G}. Since G is intersecting, Afc(F) fi G = 0 holds. Also, for G,G' G G we have \(X -G)n(X- G')| = \X\ - \G U G'| > n - 2ft + 1, i.e., F is (n - 2ft + l)-intersecting.

INTERSECTION THEOREMS FOR FINITE SETS 1423

Applying Theorem 3.3 with fc = n — ft and t = n - 2ft -j-1 gives

Using |Afc(P)| + |G| < (»), we infer |G| < (h/n){nh) = (£}) , as desired.

For a famiy F C (*) and 0 < ft < fc the ftth containment matrix M^(F) is a |A/!(F)| by |F| matrix whose rows are indexed by the members G of A/j(F), the columns by F G F and the (G, F)-entry is 1 or 0 according to whether G C F or G £ F hold.

THEOREM 3.4 (FRANKL AND FÜREDI [FF1]). Suppose that F C (*), 0 < g < k, and rank(Afff (F)) = |F| holds. Then

|Ah(P)|/ |P|> ( * £ * ) / ( * £ * ) holds for all g <h<k. (3.2)

REMARK 3.5. It can be shown that if F is t-intersecting then always

rank(M*_t(F)) = |F|

holds and therefore Theorem 3.4 is a generalization of Theorem 3.3. In this context we should mention the Kruskal-Katona theorem, one of the

most important theorems in extremal set theory. Given m, fc, and ft, this theorem describes the minimum size of Afc(F) over all families F, consisting of fc-element sets and satisfying |F| = m.

Let N be the set of positive integers. We define a total order (called the reverse lexicographic order) on (^) by setting A < B if and only if max{a: a G (A - B)} < max{&: be(B-A)}. E.g., {10,11} < {1,12}.

THEOREM 3.5 (KRUSKAL [Kr] AND KATONA [K2]). The size of the h-shadow is minimized over all F C (^) with \F\ = m, by taking the smallest m sets in the reverse lexicographic order.

REMARK 3.6. The optimal families are not always unique in the Kruskal-Katona theorem. Füredi and Griggs [FG] characterized the values of (m, fc, ft) for which there is a unique optimal family. See Mors [M] for some refinement of Theorem 3.5 and [F3] for a simple proof.

4. Families with prescribed intersections. An i-system F C ("£) is called an (n, fc, L)-system.

DEFINITION 4.1. m(n,k,L) =max{|F|: F is an (n,fc,L)-system}. Note that with this definition the Erdös-Ko-Rado theorem can be rephrased

as

m(n, fc, {*, t + 1,. . . , fc - 1}) = y, " J for n > n0(fc, t). (4.1)

1424 PETER FRANKL

THEOREM 4.2 (RAY-CHAUDHURI AND WILSON [RW]).

m(n,k,L)<^. (4.2)

For n > no(,L) the inequality (4.2) was improved in the following way. Let L = {l1,...,l9} with 0 < h < • • • < ls < fc.

THEOREM 4.3 (DEZA, ERDÖS, AND FRANKL [DEF]) . Suppose that n > no(k,L). Then

m ( n , f c , I ) < I | ^ , (4.3) leL

moreover, m(n,k,L) = 0(ns~x) unless (1% — h)\(h ~h)\ • • • \(h — ls-i)\(k — ls)-

Note that (4.3) implies (4.1). Deza [D] proves that equality is possible in (4.3) for given n, fc, ls, l3-i,..., h if and only if there exists a matroid on n vertices, of rank s + 1, in which all hyperflats have size fc and all i-flats have size Z +̂i for i = 0 , 1 , . . . , s — 1. Such a matroid is called a perfect matroid design. Examples include affine, vector, and projective spaces, their truncations, Steiner-systems, etc.

It is a tantalizing open question to decide for which (fc, L) does m(n, fc, L) > cn8 hold with some positive constant c = c(k,L).

The first open questions are

fc = 13, L = {0,1,3} and fc = 11, L = {0,1,2,3,5}.

One cannot expect to determine m(n, fc, L) in general because, e.g.,

m ( / 2 + / + l , / + l,{0,l}) = /2 + i + l

if and only if a projective plane of order I exists. In general, for L = { 0 , 1 , . . . , t — 1} an easy consideration gives

m (n,k,{0,l,...,t-l})< ( n ) / ( j f o r r c > f c > * > l

Moreover, equality holds if and only if there exists S C (£) with the property that every i-subset of X is contained in exactly one member of S (i.e., S is a (t, fc, n)-Steiner-system). For t = 1 Steiner-systems exist iff k\n. For t = 2 a celebrated result of Wilson [W2] shows that the trivial necessary conditions ((fc — l) |(n - 1) and (2)1(2)) a r e sufficient for n > rio(k). However, very little is known for t > 3. In particular, no Steiner-system is known with t > 6.

. ^Jxe ixesMower^ an ingenious^ application of the probabilistic method—is due to Rodi.

T H E O R E M 4.4 ( R ö D L [R]).

^lim^m^fc, {0,1,... ,t - l } ) / ( n ) = V U ) f°r allk^t^1' (44)

The following result is an improvement on (4.2).

INTERSECTION THEOREMS FOR FINITE SETS 1425

THEOREM 4.5 (FRANKL AND WILSON [FW]). Suppose thatp is a prime and q(x) is an integer-valued polynomial of degree d with p\q(k) but p\q(l) for all l G L. Then

\F\ < |Ad(P)| < (n] holds. (4.5)

To deduce (4.2) from (4.5) take q(x) = Y\ieL(x-I). To prove (4.5) one shows that if F satisfies the assumptions of the theorem, then Md(F) has rank equal to F. Since Md(F) has |A<j(F)| rows, this implies (4.5).

Let us recall that f(n) = Q(na) means that there exist positive constants ci and C2 such that limn_,oo inf f(n)n~a > c1 and limn_>oo sup f(n)n~a < c2

holds. It is trivial to see that m(n, fc, {/}) = &(n). It follows from Theorem 4.3

that m(n, fc, {h,fa}) = 0(n2) or 6(n) according to whether (fa — h)\(k — fa) or (fa-h)\(k-fa) holds.

For \L\ = 3 the situation is already much more complicated. In [F4] examples with m(n,k,L) — 8(n3/2) and in [FÜ3] with m(n,k,L) — 6(n4/3) are exhibited (in both cases |L| = 3). On the other hand, in [F4] it is shown that for \L\ = 3 either m(n, k,L) = 0(n3/2) or m(n,k,L) > c(k,L)n2 holds.

THEOREM 4.6 [F5]. For every rational number r > 1 there exists an infinity of choices of fc = k(r), L = L(r) such that m(n, k,L) = Q(nr) holds.

This theorem raises two problems. Problem 4.7. Are there irrational numbers a such that m(n,k,L) = &(na)

holds for some fc and Ll Problem 4.8. Given fc, L does there always exist a real number a > 1 such

that m(n,k,L) = G(na) holds?

5. Families with one forbidden intersection size. For fc > / > 0 let us introduce the notation

m(n, fc,/) = m(n, fc, {0,1, . . . , fc - 1} — {/}).

That is, m(n, fc, J) is the maximum number of fc-subsets of an n-set without two which intersect in exactly / elements.

There are two natural ways to avoid intersection size /. One is to take an (Z-hl)-intersecting family, e.g., the (]JZ{zJ) fc-element sets through a fixed (l + l)-element set. The other is to take an /-packing, i.e., a collection of sets any two of which intersect in less than / elements. This could not produce more than (?)/(?) ^-subsets. One can do better.

Let S C (2k-i-i) be an (n, 2fc —/ —1,{0,1,...,/ —l})-system of maximal size. Define F = A*(S).

By (4.4) we have

u-a-<<-i,-i)(?)/r.,~i> M

and one can check that F contains no two sets with intersection of size /.

1426 PETER FRANKL

The next result, which was conjectured by Erdös [E2], shows that for fc > 2Z+1 the first construction is best possible.

THEOREM 5.1 (FRANKL AND FÜREDI [FF2]) . Suppose that I > 0, fc > 21 + 2, and F C (*) satisfies \F n F'\ ± I for all F, F1 G F. Then

| F | < U ~ ! ~ | ) holds forn >n2(k, I). (5.2)

Moreover, equality holds in (5.2) if and only if

F = {Fe(Xk}TCF} foroomeTe^).

Clearly, Theorem 5.1 is a strengthening of the Erdös-Ko-Rado theorem, except that the bounds we have for r&2(fc, /) are rather poor. For fc < 2/ + 1 we have the following partial complement to (5.1).

THEOREM 5.2 [F6]. Suppose that F C (*) contains no two sets whose intersection has size I, ft < 21 + 1. Then

.„. ^ /2fc-/-l\ (n\ /(2k-l-l\ , f J . , , , c o X

|F| < I ) ( / ) / ( ) holds for k — l a prime power. (5.3)

Moreover, if k — I is a prime, then equality holds in (5.3) if and only if there exists an (1,2k —I — 1,nj-Steiner-system, S with F = Afc(S).

CONJECTURE 5.3. The statement of Theorem 5.2 is true for all fc and I satisfying fc < 21 + 1.

Let us mention that for fc = 21 + 1 the value of the RHS of (5.3) is (™) which

is only slightly larger (for n —• oo) than (£Z{ZÌ)-Let us also mention, that one can weaken the assumption on fc - / in Theorem

5.2 to: fc — / has a prime power divisor greater than l (cf. [FS]). Let us consider now the nonuniform case, i.e., there is no restriction on |F |

for FeF. Let us define

m(n,ì) = max{|P|: F C 2X, \F PiF'\£l for all distinct F,F' G F} .

It is clear that Ao = {A C X: \A\ < I or \A\ > (n + 0/2} has no two members intersecting in / elements. As in the case of the Katona theorem, for n+l even we can extend A 0 to Ai = Ao U (( n^) / 2 ) j Y ^ (n-i)» without losing this property.

Using Theorem 3.4, Füredi and the author proved that for n > no(l) these constructions are best possible. _ _

THEOREM 5.4 [FF1]. Suppose that I > 1 and n > n0(l). Then one of (i) and (ii) holds.

(i) n + l is odd, m(n,ì) = |Ao| and A0 is the only optimal family. (ii) n + l is even, m(n,t) = |Ai | and Ai is the only optimal family.

Let us note that our proof only gives no(l) ;S 3*. However, Conjecture 5.3 would imply no(l) ^ 6 / .

INTERSECTION THEOREMS FOR FINITE SETS 1427

For the case I ^ o(n) we have no exact results. However, Rodi and the author proved, e.g., m(n, L?T/4j) < 1.99n, i.e., it is exponentially smaller than 2n.

THEOREM 5.5 (FRANKL AND RÖDL [FRI ] ) . Suppose that a is a real number, 0 < a < \. Then there exists a positive real number e — e(a) such that

m(n,ì) < (2 — e)n holds for all integers I satisfying an <l < (\ — a)n. (5.4)

REMARK 5.6. This result was conjectured by Erdös [E2], and it appears in the list of his favorite problems, cf. [E3]. Let us note also, that Conjecture 5.3 would imply (5.4) with a better e = e(a).

For two families A and B let ij(A,B) denote the number of pairs (A,B) satisfying A G A, B G B, and \A fl B\ = /.

THEOREM 5.7 (FRANKL AND RÖDL [FRI ] ) . For every positive S there exists a positive e = e(6) such that for all integers a,b,l and families A C (*), B C (£) satisfying |A| |B| > (£) (£)(1 - e)n one has

i K A , B ) > ( l - ô ) % ( ^ , ^ ) ) . (5.5)

Clearly Theorem 5.7 is much stronger than Theorem 5.5. E.g., for / = [n/4j it implies the following.

COROLLARY 5.8. For every 6 > 0 there exists e = e(6) > 0 such that &Ln/4j(F,F) > (4 - 6)n holds for every family F C 2X and satisfying \F\ > (2-e)n.

6. Euclidean Ramsey theory. A finite subset A of Rd is called Ramsey (cf. [EG]) if for every r there exists n = no(A,r) with the property that for every partition Rn = X\ U • • • U Xr there is some i and A1 C Xi with A' congruent to A. Trivially, if \A\ — 2 then A is Ramsey with no(A,r) < r.

Solving an old problem of Hadwiger, with the help of Theorem 4.5, the fol­lowing was proved.

THEOREM 6.1 [FW]. Suppose that r < 1.2n and Rn = Xi U• • • UXr. Then there exists an i such that for every positive real 6 there are two points in Xi whose distance is exactly 6.

Let us note that Larman and Rogers [LR] showed that one cannot replace 1.2 by 3.

Erdös, Graham, Montgomery, Rothschild, Spencer, and Straus [EG] showed that the vertex set of all d-dimensional rectangular parallelepipeds is Ramsey. On the other hand they proved that if A is Ramsey, then A is spherical; i.e., it has a circumcenter, a point at equal distance to all of them. This leaves open, e.g., the case of nondegenerate, obtuse triangles. Using Ramsey's theorem Rodi and the author [FR2] proved that all nondegenerate triangles were Ramsey.

Let Sp = {(xo, •••,Xd) G Rd+1: x% H (- x\ = p2} be the d-dimensional sphere of radius p.

1428 PETER FRANKL

Let us call a set A C Rd exponentially sphere-Ramsey if there exist positive reals p = p(A) and e = e(A) such that for every partition S™ = X\ U • • • U Xr

with n > d, r < (1 + e)n, there exist an i and A' C Xi with A1 congruent to A.

THEOREM 6.2 [FRI]. Suppose that A is the vertex set of a nondegenerate simplex or of a parallelepiped in Rd. Then A is exponentially sphere-Ramsey.

REMARK 6.3. If A is a parallelepiped with circumradius 7 then one can take p(A) = 7 + 6 for an arbitrary 6 > 0. This proves, in a stronger form, a conjecture of Graham [G].

The main tool in proving Theorem 6.2 is a rather complicated extension of Theorem 5.7 to families of partitions.

One can use Corollary 5.8 to prove the following strengthened version of a conjecture of Larman and Rogers [LR].

THEOREM 6.4 [FRI ] . To every r > 2 there exists e - e(r) > 0 such that in every family of more than (2 — e)4n (±l)-vectors of length 4rc one can find r which are pairwise orthogonal.

7. Multiple intersections. Let us call F C 2X r-wise t-intersecting if |Fi n ••• fl Fr\ > t holds for all F i , . . . , F r G F. Thus 2-wise t-intersecting means simply t-intersecting. Let g(n, r, t) denote the maximum size of an r-wise t-intersecting family F C 2X, n > t. By trivial considerations g(n + l , r , t ) > 2g(n, r, t) holds. Therefore the limit p(r, t) = limn_>oo g(n, r, t)2~n exists.

Obviously, p(r, t) > 2 - t . The lower bound part of the Katona theorem shows that p(2, t) = \ for all t > 1. For r > 3 let ßr be the unique root of the polynomial xr — 2x + 1 in the interval (0,1). It is easy to see that ßr is monotone decreasing in r and it is tending to | exponentially fast.

THEOREM 7.1 [F7]. There exists a constant c = c(r) such that

cfit/y/t<p{rtt)<ft. (7.1)

The lower bound is obtained via the following construction. Let Bi G (t+ri) and define B* = {F C X: \F n B{\ > t + (r - l)i}. Clearly, B* is r-wise t-intersecting.

CONJECTURE 7.2 [F8].

i

Let us note that in [F8] (7.2) is proved for t < 2 rr/150. In [F7] it is proved for r = 3, t = 2, giving g(n, 3,2) = 2n~2.

For the proof of these results a shifting technique, introduced by Erdös, Ko, and Rado, plays a crucial role. Let us mention that Kalai [Ka] defined a more powerful, algebraic shifting, which proved very useful in various situations.

INTERSECTION THEOREMS FOR FINITE SETS 1429

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