3-Wise Exactly 1-Intersecting Families of Sets

Graphs and Combinatorics (2005) 21:71–76 Digital Object Identifier (DOI) 10.1007/s00373-004-0592-x

Graphs and Combinatorics Ó Springer-Verlag 2005

3-Wise Exactly 1-Intersecting Families of Sets Zsolt Katona Department of Probability e-mail: [email protected]

Theory

and

Statistics,

Eo¨tvo¨s

Lora´nd

University.

Abstract. Let f(l, t, n) be the maximal size of a family F
2½n such that any l  2 sets of F have an exactly t  1-element intersection. If l  3, it trivially comes from [8] that the optimal families are trivially intersecting (there is a t-element core contained
by all the members of the family). Hence it is easy to determine f ðl; t; nÞ ¼ 2l ðn  1Þ þ 1. Let gðl; t; nÞ be the maximal size of an l-wise exaclty t-intersecting family that is not trivially t-intersecting. We give upper and lower bounds which only meet in the following case: gð3; 1; nÞ ¼ n2=3 ð1 þ oð1ÞÞ. Key words. Extremal problems for families of finite sets, Finite projective geometries

1. Introduction   Let ½n denote the set f1; 2; . . . ; ng, 2½n the collection of all subsets of ½n and ½n k the set of k-element subsets of ½n. A simple theorem of Erd} os, Ko and Rado [6] says that the maximum of jFj is 2n1 , if every two members of a family F  2½n have a non-empty intersection. Such an F is called an intersecting family. A maximum intersecting family F can be obtained by considering all the subsets containing one fixed element. This F sometimes is called a cylinder, or trivially intersecting, since \ F :¼ \F 2F F 6¼ ;. A family F
2½n (consisting of distinct sets) is called l-wise exactly t-intersecting if jF1 \    \ Fl j ¼ t holds for any distinct members F1 ; . . . ; Fl of F and is called trivially l-wise exactly t-intersecting if j \ Fj ¼ t. Let f ðl; t; nÞ denote the maximal size of an l-wise exactly t-intersecting family F
2½n . The well-known theorem of Erd} os and De Brujin [5] states that f ð2; 1; nÞ ¼ n. There are three types of optimal families here. One is trivially intersecting: ff1gg [ ff1; igjn  i  2g, another one is almost the same: f½n  f1gg[ ff1; igjn  i  2g, but the third one is PGð2; qÞ ( if q2 þ q þ 1 ¼ n). The generalization for t  2 is known as Fisher’s inequality. R.C. Bose was the first to apply the linear argument to solve a combinatorial problem [2], but the following theorem was first proved by Majumdar.

AMS classification: 05D05; 05B25

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Theorem. [15] If t  1 then f ð2; t; nÞ  n. We are interested in the case l  3. The general problem of r-wise at least s-intersecting and l-wise (l  r  2) at most t-intersecting families is solved in [9] only for t  2s  2. To determine f ðl; t; nÞ is a special case of the general problem (t ¼ s, r ¼ l) not included in [9]. It easily comes from a theorem of Fu¨redi [8] that the optimal families here are trivially intersecting. Another approach is the linear algebraic method. The well known, far reaching generalizations of the Erd} os-de Bruijn theorem are the results of RayChaudhuri-Wilson [16] and Frankl-Wilson [7] about 2-intersections. Continuing this way Grolmusz and Sudakov [11] gives upper bounds for families with given l-intersections. The referee called my attention to two recent papers in the field. Fu¨redi and Sudakov in [10] and Szabo´ and Vu in [18] also study families with given l-intersections and they prove more general theorems than the following. For the sake of completeness, in Section 2, we determine the exact value of f ðl; t; nÞ, proving this theorem.
l  Theorem 1. Suppose l  3 and n  lðlþtÞ l2 , then f ðl; t; nÞ ¼ 2 ðn  tÞ þ 1. There are several examples for optimal families that are not trivially intersecting. The finite projective plane is one of these which is non-trivially 2-wise exactly 1-intersecting. Also in [14] where intersecting but l-wise at most 1-intersecting families are studied, the only asymptotically optimal family is originated from the finite projective plane. This leads us to examine the maximal non-trivially l-wise exactly t-intersecting families in Sections 3 and 4. Let gðl; t; nÞ ¼maxjFj where F is l-wise exactly t-intersecting and j \ Fj < t. Of course gðl; t; nÞ  f ðl; t; nÞ. By [5] gð2; 1; nÞ ¼ n. In Section 3 we prove the upper bounds, in Section 4 the lower bounds of the following theorems. Theorem 2. gð3; 1; nÞ ¼ n2=3 ð1 þ oð1ÞÞ: In general, our lower and upper estimates are, unfortunately, rather different. Theorem 3. If l  2; then 1 1=l n ð1 þ oð1ÞÞ  gðl; t; nÞ  nðtþ1Þ=ðlþt1Þ ð1 þ oð1ÞÞ: t These theorem yield too, that the optimal families in Theorem 1 are trivially intersecting for large n’s. 2. Trival Intersection Allowed Proof of Theorem 1. A special case of a theorem of Fu¨redi says Theorem 8. If l  3; t  1 and F is l-wise non-trivially exactly t-intersecting then jFj  n þ l  2.

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A much stronger theorem (Theorem 3) will be proved in Section 3 but we wanted We will construct a family with
to keep  the linear ordering of the paper. l size 2l ðn  tÞ þ 1 > n þ l  2 (for n  2 þ l2 t) if j \ Fj ¼ t so we only have to deal with the trivially intersecting case. (Suppose that \F ¼ fn; n  1; :::; n  t þ 1g. Define the family F0 ¼ fF fn; n  1; :::; n  tþ1g jF 2 Fg. Trivially jFj ¼ jF0 j. For every F 0 2 F0 let us associate a weight jF10 j with every element of F 0 except the empty set. Since every element of f1; 2; . . . ; n  tg is contained in at most l  1 subsets we have at most l  1 weights on every element. The two largest weights cannot be both 1s because all subsets are different and two 1s would mean two different 1-element subsets. The second largest possible l weight is 12, so the sum of the weigths is at most 1 þ l2 2 ¼ 2 on every element. The l sum of the weigths on all elements is at most ðn  tÞ 2, and is equal to jF0 j or jF0 j  1 (depending on the empty set) because the sum of the weigths is 1 for each subset. Thus l ð1Þ jFj ¼ jF0 j  ðn  tÞ þ 1: 2
 If n  l þ t we can construct an F with jFj ¼ 2l ðn  tÞ þ 1. It is enough to   exhibit an F0 with jF0 j ¼ 2l ðn  tÞ þ 1 on the underlying set f1; 2; . . . ; n  tg. In the case of even l let F0 be ; [ f1g [ f2g [    [ fn  tg [ F01 [ F02 [    [ F0l2 2 where F0i ¼ f1; i þ 1g [ f2; i þ 2g [    [ fn  t; n  t þ ig (every element is 0 understood n  t). In the case of
odd    [  l let F
ntbe  ; [ f1g[f2g
[ 0 0 0 nt nt [ f1; fn  tg [ F [ F [    [ F þ 1g [ f2; þ 2g [    [ f ; l2 1 2 2 2 2
 b2c 2 nt h 2 g.

3. Upper Bounds for the Non-Trivial Case Here we prove the upper bounds of Theorem 2 and 3 by induction on l. We will show jFj  nðtþ1Þ=ðlþt1Þ ð1 þ oð1ÞÞ if F is l-wise exactly t-intersecting (l  2) and j \ Fj < t. The first case, l ¼ 2 is Fisher’s inequailty. Before proving the induction step we have to remark that the upper bound is true and the proof is valid if we allow multisets in F (if it is a collection of not necessary different subsets of 2½n ) and we will use this fact in the induction step. Suppose for l  3 that the bound is true for l  1 and consider a non-trivially l-wise exactly t-intersecting family F. Denote the smallest member of F by X and suppose jX j ¼ k. Take the intersections X \ F for all F 2 F. Consider them as a collection of not necessary different subsets of X as mentioned above. One can see that these intersections are non-trivially l  1-wise exactly t-intersecting (j \F 2F X \ F j < t). By induction this implies jFj  k ðtþ1Þ=ðlþt2Þ ð1 þ oð1ÞÞ:

ð2Þ

Hence we are done with the proof if k  nðlþt2Þ=ðlþt1Þ so we can suppose  k > nðlþt2Þ=ðlþt1Þ . Before continuing the proof extend the notation xt for

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 non-integer x’s. Let xl ¼ f ðxÞ ¼ xðx1Þðxlþ1Þ if x  l  1 and f ðxÞ ¼ 0 otherwise. l! One can see that this function   is monotone and convex so we can apply Jensen’s inequality. For an A 2 ½nt let dA denote jfF 2 FjA
F gj, the number of sets of F containing A. The l-wise exactly t-intersecting condition means that for every l sets in F there is a set jAj ¼ t which is their intersection. A can be the intersection of dlA choices of l members of F if dA  l. Hence 



jFj l n t

P ¼

A2ð

Þ n ½n t

d 

t

A

l

0 @

1

P

A dA A ðntÞ

ð3Þ

l

by the Jensen-inequality. Fora fixed A there are dA F ’s (2 F) containing A and for a fixed F 2 F there are jFt j A’s in it, so we can continue (3): 0P

A n t

dA

1

0P

@ ðÞ A¼@ l

1 ðjFt jÞ1 0 ðk Þ jFj nt ðÞ A@ ð t Þ A: l l

F 2F n t

ð4Þ

we would be done with the proof. We may suppose jFj > nðtþ1Þ=ðlþt1Þ , otherwise ðk Þ Since k > nðlþt2Þ=ðlþt1Þ the quantity jFj nt tends to infinity (n ! 1). The ðtÞ combination of (3) and (4) leads to 0 1 !l k  k!l ðktÞ jFj n 1 1 1e l t ð Þ @ A t n jFj  ðjFj n  l þ 1  jFj nt   l! l! l! l t t t

ð5Þ

for any e > 0 if n is large enough. Hence we obtain nl1 k l  ð1 þ oð1ÞÞ t t

ð6Þ

and this yields k  ð1 þ oð1ÞÞnðl1Þ=l  ð1 þ oð1ÞÞnðlþt2Þ=ðlþt1Þ , so by (2) jFj  ð1 þ oð1ÞÞntþ1=ðlþt1Þ . h 4. Non-Trivial Constructions Constructions will prove the lower bounds of Theorem 2 and 3. We will use projective spaces, namely PGðN ; qÞ, the N -dimensional finite projective space of order q. It consists of qN þ qN 1 þ    þ 1 hyperplanes on the same number of points. Considering the hyperplanes as sets on the underlying points, any N of them have a non-empty intersection, which can be a K-dimensional subspace K < N  1. First we show gð3; 1; nÞ  n2=3 ð1 þ oð1ÞÞ to complete the proof of Theorem 2 giving a family F consisting of n2=3 ð1 þ oð1ÞÞ subsets of ½n which are 3-wise

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1-intersecting, and \ F ¼ ;. Take PGð3; qÞ, such that q3 þ q2 þ q þ 1 ¼ n (if there is such prime power q). By [2] and [17] the maximal size of an ovoid is q2 þ 1 here, which means that the maximal number of points, such that there at most two on any line, is q2 þ 1. Take such q2 þ 1 points. Obviously, these points can not lie in one plane. Their dual planes (there are also q2 þ 1 of them) will be the members of F. They have the property that no three of them contain the same line. This implies that the intersection of 3 planes is a point, so F is 3-wise 1-intersecting, and of course, non-trivially. Hence gð3; 1; nÞ  n2=3 ð1 þ oð1ÞÞ because the underlying set has q3 þ q2 þ q þ 1 points. If there is no q such that q3 þ q2 þ q þ 1 ¼ n and q is prime power, then we take the largest m  n such that q3 þ q2 þ q þ 1 ¼ m with a prime power q. We know that there is a prime between n and ð1  eÞn for any e if n is large enough. To prove gðl; 1; nÞ  n1=l ð1 þ oð1ÞÞ for l  4 (Theorem 3) we generalize the above construction. We would like to find as many points as we can in PGðl; qÞ (ql þ    þ q þ 1 ¼ n) such that there are at most l  1 of them in any l  2-dimensional (2-codimensional) subspaces (these points are called a k-track, if there are k of them). Unfortunately the exact number of these points is not pffiffiffi known for l  4, but we have a lower bound q þ 2 q ([4] and [1]). Taking these pffiffiffi q þ 2 q points, their dual hyperplanes will form F, that will be l-wise 1-intersecting as above and so gðl; 1; nÞ  n1=l ð1 þ oð1ÞÞ. Finally gðl; t; nÞ  ð1=tÞn1=l ð1 þ oð1ÞÞ comes with replacing every element of the underlying set with t other ones.

Remark 1. If the exact value of the size of a track was q2 ð1 þ oð1ÞÞ (this is the existing upper bound) this would imply gðl; 1; nÞ ¼ n2=l ð1 þ oð1ÞÞ with the idea above. The reader may be interested in arcs, caps, ovoids and (n,r)-sets. All recent results on them can be found in the survey [12] and its update [13].

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9. Fu¨redi, Z., Katona, Zs.: Multiply intersecting families of sets, J. Comb. Theory, Ser. A, 106/2, 315–326 (2004) 10. Fu¨redi Z., Sudakov, B.: Extremal set-systems with restricted k-wise intersections, J of Comb. Theory, Ser. A, Vol 105, 143–159 (2004) 11. Grolmusz, V., Sudakov, B.: k-wise Set-intersections and k-wise Hamming-distances, J. Comb. Theory, Ser. A 99, 180–190. (2002) 12. P Hirschfeld, J.W., Storme, L.: The packing problem in statistics, coding theory and finite projective spaces. R. C. Bose Memorial Conference (Fort Collins, CO, 1995) J. Stat. Plann. Inference 72, 355–380 (1998) 13. P Hirschfeld, J.W., Storme, L.: The packing problem in statistics, coding theory and finite projective spaces: update 2001 http://cage.rug.ac.be/~ls/ 14. Katona, Zs.: Intersecting families of sets, no l containing two common elements, Discrete Math. 226, 233–241 (2001) 15. Majumdar, K.N.: On some theorems in combinatorics relating to incomplete block designs, Ann. Math. Stat. 24, 377–389 (1953) 16. Ray-Chaudhuri, D.K., Wilson, R.M.: On t-designs, Osaka J. Math. 12, 735–744 (1975) 17. Qvist, B.: Some remarks concerning curves of the second degree in a finite plane, Ann. Acad. Sci. Fenn., Ser. A. 134 (1952) 18. Szabo´ T, Vu, V.: k-wise intersection theorems, submitted. Download from (http:// www.math.ucsd.edu/vanvu/papers/algebraic.html)

Received: June 11, 2003 Final version received: July 15, 2004