Exact k-wise intersection theorems

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1 Exact k-wse ntersecton theorems Tbor Szabó V. H. Vu Abstract A typcal problem n extremal combnatorcs s the followng. Gven a large number n and a set L, fnd the maxmum cardnalty of a famly of subsets of a ground set of n elements such that the ntersecton of any two subsets has cardnalty n L. We nvestgate the generalzaton of ths problem, where ntersectons of more than 2 subsets are consdered. In partcular, we prove that when k 1 s a power of 2, the sze of the extremal k-wse oddtown famly s (k 1(n 2 log 2 (k 1. Tght bounds are also found n several other basc cases. 1 Introducton In ths paper we study famles F of subsets of an n-element ground set [n] = {1, 2,..., n}, for whch the sze of the ntersecton of any k members of F s n a prescrbed set of ntegers. On the most general level our problem can be formulated as follows. Gven a postve nteger n, a set L of ntegers, and an nteger k = k(n > 1 fnd the maxmum cardnalty of a famly of subsets of [n] such that the ntersecton of any k of these subsets has a cardnalty contaned n L. For k = 2, these types of questons have been attacked successfully usng lnear algebrac methods. The excellent monograph [2] by Baba and Frankl contans a wde varety of results and applcatons. In the followng we lst some of the most basc theorems. Sets of even (odd cardnalty are called even (odd. A condton on the k-wse ntersecton s always about k parwse dstnct members of the famly. Eventown Theorem (Berlekamp [5], Graver [11] Let F 2 [n] be a famly such that the ntersecton of any two subsets n F s even. Then F 2 n/2 + δ n, where δ n = 1 f n s odd and 0 otherwse. Department of Computer Scence, ETH Zürch, 8092 Swtzerland. Emal address: szabo@nf.ethz.ch. Research supported n part by NSF grant DMS and by the jont Berln/Zurch graduate program Combnatorcs, Geometry, Computaton, fnanced by the German Scence Foundaton (DFG and ETH Zürch. Department of Mathematcs, UCSD, La Jolla, CA , USA. Emal address: vanvu@ucsd.edu; webpage:, vanvu/ Research supported n part by NSF grant DMS , by an NSF CAREER award and by an Alfred P. Sloan fellowshp. 1

2 Oddtown Theorem (Berlekamp [5] Let F 2 [n] be a famly of odd subsets, such that the ntersecton of any two subsets n F s even. Then F n. Nonunform Fscher Inequalty (Majumdar [14] Let F 2 [n] be a famly such that the ntersecton of any two subsets n F has the same nonzero cardnalty. Then F n. Nonunform Ray-Chaudhur-Wlson Theorem (Frankl and Wlson [6] Gven a subset L of s nonnegatve ntegers, let F 2 [n] be a famly such that the ntersecton of any two members of F has a cardnalty contaned n L. Then F s ( n. It s mportant to note that all the above bounds are best possble. The generalzaton of these theorems from parwse to k-wse ntersectons seems completely natural, stll the adaptaton of the arguments poses some nontrval challenge. The reason for ths dffculty les exactly where the beauty of the proofs for k = 2 s. Arguments about parwse ntersectons usually depend on the basc fact that the sze of the ntersecton of two sets s equal to the nner product of the two correspondng characterstc vectors, hence the machnery of lnear algebra can be nvoked. The ntrnsc dffculty of the k-wse case compared to the parwse s the lack of algebrac concepts descrbng the ntersecton of k sets when k > 2. In ths paper we try to crcumvent ths obstacle wth addtonal combnatoral deas whle stll makng some use of lnear algebra. The general problem of nvestgatng k-wse ntersecton restrctons on famles of sets for k was posed n [15] by V. T. Sós. Füred [8] has establshed a combnatoral connecton between the parwse and k-wse case for t-unform systems. He showed that for any fxed L, the order of magntude of the sze of the extremal system for the parwse and k-wse problem s the same. Hs constants are very large, but depend only on k and t. The k-wse varant of specfc ntersecton problems were studed recently by the second author [16, 17], Grolmusz [12], and Grolmusz and Sudakov [1]. An mportant feature of the lnear algebra method s that the obtaned results are often tght. There are, usually qute smple, matchng constructons complementng the algebrac upper bounds; not just up to a constant factor or asymptotcally, but precsely. Sharp results, just because of ther scarcty, are of partcular nterest n Extremal Combnatorcs. In ths paper we put the emphass on obtanng precse results for the k-wse verson of some of the classcal parwse ntersecton theorems, lke the Oddtown Theorem, or the Nonunform Fscher Inequalty. 1.1 k-wse results A precse bound for the k-wse verson of the Eventown Theorem was gven by the second author n an earler paper [16]. Here we quote ths result for the sake of completeness. Theorem 1.1 There s a postve constant c such that the followng holds. If n c log 2 k and F 2 [n] s a famly of maxmum sze such that the ntersecton of every k subsets n F s even, then 2

3 F = 2 n/2 + δ k,n, where δ k,n = k 1 f n s odd and 0 otherwse. We call a famly F 2 [n] k-wse oddtown, f every member of F s odd and the ntersecton of any k members s even. The followng s conjectured to be the optmal famly f n s large enough compared to k. Constructon A. [16] Let us consder b = log 2 (k 1 dsjont 2-element subsets of [n], say X = {2 1, 2} for = 1,..., b. By takng the unons of some of these pars, create k 1 sets Y 1,..., Y k 1 ; say let Y := X j1... X jr, where j 1,..., j r are the nonzero coordnates n the bnary expanson of 1. Defne F = {Y {x} : = 1,..., k 1; x = 2b + 1,..., n}. Then F = (k 1(n 2 log 2 (k 1 and F s k-wse oddtown. Indeed, no k-wse ntersecton contans anythng from {2b + 1,..., n}, whle ponts from [2b] come n pars. In the followng theorem we confrm that for an nfnte sequence of k Constructon A s, ndeed, extremal. We also obtan an almost tght bound for every k. Theorem 1.2 Let k = k(n be an nteger and F 2 [n] be a k-wse oddtown famly of maxmum sze. If k = 2 s + 1 for some nteger s, then for every n k 1 k 2 log 2(k 1 F = (k 1(n 2 log 2 (k 1. Moreover, for every k and n (k log 2 (k 1 2, (k 1(n 2 log 2 (k 1 F (k 1(n log 2 (k 1 log 2 (k 1. We consder Theorem 1.2 the man contrbuton of our paper. Prevously the asymptotcs of the sze of the maxmum famly was establshed [16] for fxed k, but exact results were not known for any k > 2. We also prove the followng precse k-wse verson of the Nonunform Ray-Chaudhur-Wlson Theorem for L = 1. Theorem 1. Let F 2 [n] be a largest famly such that the ntersectons of any k = k(n subsets n F has the same cardnalty. If k 1 > 2 n 1, then F = 2 n. Otherwse let s be the largest nteger such that s 1 k 1. Then ( n 1 F = s ( ( n + k 1 s 1 ( n 1 n s + 1. (1 In partcular for n k, we have F = k 2 n + 1.

4 Ths theorem mproves on the earler bound of (k 1(n + 1 of Grolmusz and Sudakov [1]. For parwse ntersectons, the Nonunform Ray-Chaudhur-Wlson Theorem s sharp only when L = {0}. In case L {0}, the Nonunform Fscher Inequalty mproves the upper bound n + 1 to n. A smlar phenomenon occurs here as well: Theorem 1. s only sharp f all k-wse ntersectons are empty. The followng statement s a k-wse varant of the Nonunform Fscher Inequalty. Theorem 1.4 Assume that F 2 X k = k(n members of F has the same nonzero cardnalty. Then { f(k, n 1 f k 1 2 n 2 F = k 1 otherwse where f(k, n s the quantty on the rght hand sde of (1. s a maxmum sze famly such that the ntersecton of any Further questons one may want to consder are the k-wse varants of the famous modular ntersecton theorems of Frankl and Wlson. The unform verson could be consdered a far-reachng generalzaton of the Oddtown Theorem. Here we dscuss the nonunform case. Nonunform Frankl-Wlson Theorem [6] Let p be a prme and L be the set of s resdues modulo p. If F 2 [n] s a famly such that for every A F, A / L (mod p and for every A, B F, A B L (mod p, then F s ( n. In a recent paper [1], Grolmusz and Sudakov proved the upper bound (k 1 s ( n for the k-wse verson of the above statement. Usng an dea from the proof of Theorem 1.1, we can slghtly mprove ther bound. Proposton 1.5 Let p be a prme and L be a set of s resdues modulo p. Suppose F 2 [n] s a famly, such that for every A F, A / L (mod p and for every A 1,..., A k F, k I=1 A L (mod p. If s ( n logp (k 1, then ( s ( n F (k 1 log p (k The rest of the paper s organzed as follows. In the next secton, we descrbe some useful deas for the proof of Theorem 1.2 and prove Proposton 1.5. The proof of Theorem 1.2 follows n Secton. In Secton 4 we prove Theorems 1. and 1.4, whle the last secton contans several concludng remarks and open questons. Notaton. The degree deg F (x of a pont x n a famly F s the number of members of F contanng x. 4

5 In Sectons 2 and we use the language of lnear algebra. For each subset A of [n], the characterstc vector of A s the bnary vector v A of length n, whose th coordnate s 1 f belongs to A and 0 otherwse. The vector space generated by the characterstc vectors of the elements of F over the two-element feld GF (2 s denoted by V (F. 2 Warm-up In ths secton, we descrbe some deas whch wll be used n the proof of Theorem 1.2. Frst we dscuss a weaker upper bound, proved by the second author as a lemma to the proof of Theorem 1.1 ([16, Lemma 6]; see also [17, Lemma.2.4] for a more general statement. Then, we modfy the proof of ths weaker bound to prove Proposton A weaker bound for the Oddtown problem Lemma 2.1 [16] If F 2 [n] s a k-wse oddtown famly and dm V (F = d > log 2 (k 1 then, F (k 1(d log 2 (k 1. Proof. We use nducton on d. The base case d = log 2 (k s mmedate, snce n any fnte dmensonal vector space over GF (2 at most half of the vectors have odd weght, thus F 1 2 V (F = 2d 1 (k 1. Assume now that d > log 2 (k Let t be the largest nteger such that there are t members A 1,..., A t of F havng an odd ntersecton A = t =1 A. Clearly t k 1. Then for every = 1,..., t, v A v A = A A = A 1 (mod 2. Due to the maxmalty of t, for any set B F = F\{A 1,..., A t } v B v A = B A 0 (mod 2. It thus follows that the vectors v A are not contaned n the space V (F generated by the famly F = F\{A 1,..., A t }. Thus the dmenson of V (F d 1 and ths, together wth the nducton hypothess, yelds F = t + F (k 1 + (k 1((d 1 log 2 (k 1 = (k 1(d log 2 (k 1, completng the proof. Corollary 2.2 [16] Let F be a k-wse oddtown famly on n > log 2 (k 1 ponts. Then F (k 1(n log 2 (k 1. 5

6 Proof. It s clear that dm V (F n. The consderaton of the dmenson of V (F turns out to be a very useful concept and wll play a crucal role n the proof of Theorem 1.2. But frst let us use the prevous dea to show Proposton Proof of Proposton 1.5 Consder a famly F as n Proposton 1.5. We mmc the method of Alon, Baba and Suzuk [1] for the proof of the Frankl-Wlson Theorem and combne t wth the deas n the prevous argument. Let W be the vector space of all polynomals generated by the set { S x S {1, 2,..., n}, S s} over the feld GF (p. It s easy to see that the generatng set forms a bass as well, thus the dmenson of W s s ( n. To each set A X, assgn a polynomal fa (x = l L (x v A l, where x denotes a vector n GF n (p. Let f A (x be the multlnear polynomal we obtan from f A (x by replacng all powers x q by the frst power x. Clearly f A (y = f A (y for any 0-1-vector y. Let d denote the dmenson of the subspace (of W spanned by the polynomals f B, B F. We shall prove, by nducton on d, that f d log p (k 1, then F (k 1(d log p (k As d dm(w = s ( n, our clam follows. Let d = log p (k 1. A subspace of W wth dmenson d has p d vectors, so t follows that F p d (k 1, concludng the proof of the base case. Now assume d > log p (k 1. Smlar to the prevous proof, let t be the largest nteger such that there are t members of F whose ntersecton has cardnalty not n L modulo p. Let these members be A 1,..., A t and let A = t =1 A. It follows that for every = 1,..., t, f A (v A = f A (v A = l L ( A A l = l L( A l 0 (mod p. On the other hand, due to the maxmalty of t, for any member B F = F\{A 1,..., A t }, B ( t =1 A = B A L. Therefore, f B (v A = f B (v A = l L( B A l 0 (mod p. Hence for every, 1 t, f A W s lnearly ndependent from the subspace V spanned by the f B, B F. Thus dm V d 1 and we can conclude as n the prevous proof. 6

7 k-wse Oddtown In order to prove Theorem 1.2, we need to consder a more general scenaro, where F s allowed to contan sets wth hgh multplcty. Ths extenson wll provde room for us to consder a more general nducton hypothess. To ths end we need to ntroduce a few defntons. A collecton F of sets s called an l-mult-system f each set occurs n F wth multplcty at most l. A 1-mult-system s called a famly. We extend the defnton of the k-wse oddtown property to mult-systems. A mult-system F s sad to have the k-wse oddtown property f A 1 (mod 2 for every A F and A 1... A k 0 (mod 2 for every sub-mult-system {A 1,..., A k } of F. That s, n the ntersecton property, A and A j mght be equal as sets, but they are dstnct members of the mult-system. Note that n a k-wse oddtown mult-system there cannot be a set wth multplcty larger than k 1. Proof of Theorem 1.2. Frst we shall prove that for an extremal k-wse oddtown system F, dm V (F should be at most n log 2 (k 1 and then apply Lemma 2.1. The followng defnton plays a crucal role n our argument. Defnton Let F be a mult-system of sets. A set M s called a bass set of F, f ( M s nonempty ( M s even ( M = A 1... A k 1 for some sub-mult-system {A 1,..., A k 1 } of F. Observe, that the characterstc vector w of a bass set M = A 1... A k 1 s orthogonal to V (F. Indeed, for any B F, B A, = 1,..., k 1, w v B = 0 snce F s a k-wse oddtown system, whle w v A = 0 because of (. Thus the vector space W (F, generated by the characterstc vectors of the bass sets of F, s contaned n the orthogonal complement V (F of V (F. It s well-known that for any subspace V, dm V + dm V = n. Therefore d n b, where b = dm W (F. In the followng theorem we prove that the maxmum sze of a k-wse oddtown famly wth dm W (F < log 2 (k 1 s asymptotcally much less than (k 1(n 2 log 2 (k 1, the lower bound gven by Constructon A. Theorem.1 Let k, l, n 1 be ntegers and F 2 [n] be a k-wse oddtown l-mult-system. Let b k 2 be the dmenson of the vector space W (F generated by the characterstc vectors of the bass sets of F. Then F { max b 2 +1 } l + k 1 (n 2. (2 Proof of Theorem.1 We proceed by nducton on b. To check the base case b = 0, assume that F s a famly wth no bass sets, that s all (k 1-wse ntersectons are ether odd or empty. We 7

8 use nducton on n. For n = 1, the maxmum sze system contans the sngleton mn{l, k 1} (2l + k 1/-tmes. Suppose now that n > 1. Case 1. Every pont s contaned n at most k 1 sets. Then A = 1 = A F p A p A F p A F 1 n(k 1. The number of sets n F s maxmzed when (almost all degrees are k 1 and the szes of the sets are as small as possble. That s when F contans all the 1-element subsets wth multplcty mn{l, k 1} and a -unform (almost-regular hypergraph wth maxmum degree (k 1 mn{l, k 1}. 1 Then F mn{l, k 1}n + (k 1 mn{l, k 1}n/ (2l + k 1n/. Case 2. There s a pont x wth degree at least k. Then there are some A 1,..., A k, such that ther ntersecton contans x. Snce ths ntersecton s even, t must contan another element y x. But A 1... A k 1 s then not empty, thus t s odd (remember that b = 0, so N = A 1... A k 2 contans at least three elements. Consder those members E of F (E A, = 1,..., k 2 for whch E N s not empty, thus odd. There are at most N of such E s. (For any two E 1, E 2, N E 1 E 2 s even. Ths mples that an odd ntersecton N E determnes E unquely. Then one can use the parwse Oddtown Theorem wthn N for the sets N E, when ths ntersecton s odd. All other members of F are dsjont from N, thus they form a system on n N ponts. So by nducton F = 2l + k 1 snce N and l 1. (2l + k 1(n N n N + N + k 2 = 2l + k 1 (2l + k 4 + k 2 n, Ths completes the proof of the base case b = 0. Now let us assume that b > 0. Let us choose a mnmal bass set M = A 1... A k 1. If M = 2, then by the k-wse oddtown property each element of F ether contans or s dsjont from M. Then the mult-system F \ M = {E \ M : E F} s a k-wse oddtown 2l-mult-system on n 2 ponts. The dmenson of W (F \ M s b 1. (More precsely, W (F s equal to the drect product W (F \ M v M. If there s a bass set B of F \ M whch s not a bass set of F, then B M s a bass set of F, thus the characterstc vector of B s n W (F. On the other hand f B s bass set of F then B \ M s a bass set of F \ M. Thus, by the nducton hypothess, { F = F \ M max b l + k 1 } ((n 2 2 { = max b 2 j+1 } l + k 1 (n 2j. j=1 1 Remark: In case k `n 1 +2, 2 Baranya s Theorem ensures that ths actually can be acheved, such that, dependng on the remander of n modulo, only 0, 1 or 2 ponts have degree smaller than the maxmum. 8

9 Suppose now that M 4. Case 1. There s an ndex j such that j A \ A j =. Then the (k 2-wse ntersecton N = j A would be equal to M and would not contan any other bass set because of the mnmalty of M. Thus, agan, every member of F ether contans or s dsjont from M, mplyng that F \ M s a k-wse 2l-mult-system. Smlarly, by the nducton hypothess, { } { } F = F \ M max b l + k 1 ((n M 2 max b 2 j+1 l + k 1 (n 2j. j=1 Case 2. For every ndex j = 1,..., k 1., the sets C j = j A \ A j are not empty. Let us choose some k 2 sets from {A 1,... A k 1 } such that ther ntersecton does not contan any other bass set, but M. Ths s possble because of a smple countng argument. We say that a bass set B M runs an ndex j f B j A. In case ndex j s runed, let us fx an arbtrary bass set B j runng j. Then the mnmalty of M mples B j C j. Snce the nonempty C j s are parwse dsjont by defnton, the set of characterstc vectors {v Bj : j s runed} s lnearly ndependent. Hence b, the dmenson of the space of the characterstc vectors of bass sets, cannot be less than (actually t s equal to the number of ndces runed. Snce b k 2, we have an ndex, say k 1, whch s not runed by any bass set. Then N = k 2 =1 A does not contan any other bass set, but M. In partcular, the ntersecton of any member of F \ {A 1,..., A k 2 } wth N s ether odd, or M, or empty. Let F N = {A 1,..., A k 2 }, F 0 = {E F \ F N : E N = }, F 1 = {E F \ F N : E N 1 (mod 2}, F 2 = {E F \ F N : 0 E N 0 (mod 2} = {E F \ F N : E N = M}. The famly F 1 = {E N : E F 1} s a parwse oddtown famly on N ponts, so F 1 = F 1 N. Any element E of F 0 F 2 s ether dsjont from N or ntersects t n M. Then (F 0 F 2 \ N = {E \ N : E F 0 F 2 } s a k-wse oddtown 2l-mult-system on n N ponts. Let the dmenson of W ((F 0 F 2 \ N be b. If B s a bass set of (F 0 F 2 \ N then ether B tself or B M s a bass set of F. In any case the characterstc vector v B s n W (F. Hence W ((F 0 F 2 \ M W (F. 9

10 Thus b b 1, snce v M W (F \ W ((F 0 F 2 \ M. By the nducton hypothess, F = F N + F 1 + F 0 F 2 k 2 + N + b max = b max b max = b max b max { l + k 1 } (n N 2 = } + 2+ l + 2k 2 + k 6 } = { 2 +2 l + k 1 (n 2( + 1 N 2+2 l + k 4 { 2 +2 l + k 1 (n 2( l + k l + 5k 8 { 2 +2 l + k 1 (n 2( + 1 } 2+2 l 12 { 2 +2 } { l + k 1 2 j+1 } l + k 1 (n 2( + 1 (n 2j b max j=1 Here we used that N 5, snce M 4 and C j. The proof of Theorem 1.2 s now mmedate by applyng Theorem.1 wth l = 1. Suppose frst that k 1 = 2 s for some postve nteger s. The sze of the famly from Constructon A s larger than the upper bound n (2 for suffcently large n. Indeed, (k 1(n 2s s larger than 2b+1 +k 1 (n 2b for every b < s f n log 2 (k 1 k 1 k 2. Hence for an extremal system F, the dmenson of W (F cannot be less than log 2 (k 1, whch mples that dm V (F n log 2 (k 1. Then the frst part of Theorem 1.2 s a consequence of Lemma 2.1. Assume now that 2 s + 1 k 1 < 2 s+1. Our argument s smlar to the above, except the range of valdty s somewhat smaller. For any n (k log 2 (k 1 2, the sze of the famly from Constructon A s greater than the domnatng term 2s+1 +k 1 (n 2s n the upper bound (2. So, agan, dm V (F n log 2 (k 1 and the second part of Theorem 1.2 follows from Lemma k-wse Nonunform Fscher-nequalty In ths secton we prove Theorems 1. and 1.4. It s convenent to ntroduce the followng defnton. We say that F s a k-wse l-fscher famly f the ntersecton of any k members of F contans exactly l elements. Let m k l (n be the largest possble sze of a k-wse l-fscher setsystem. Frst we determne m k 0 (n exactly for every n and k. Lemma 4.1 m k 0(n = { s ( n + (k 1 s 1 ( n 1 n s+1 f k 1 2 n 1 2 n f k 1 > 2 n 1, where s s the largest nteger such that s 1 ( n 1 k 1 (provded s exsts,.e. k 1 2 n 1. 10

11 Proof. For a famly H let us denote by D(H the sum of the degrees: D(H := deg H (v = A. v V A H The heart of the proof s the followng trval observaton. Observaton. A famly s k-wse 0-Fscher f and only f each pont has degree at most k 1. For k 1 2 n 1 the lemma follows easly from our observaton snce the degree of any pont n 2 [n] s 2 n 1 k 1. Suppose now that s 1 ( n 1 k 1 < s ( n 1. Frst we construct an approprate famly F of the desred sze. Let us denote by ( [n] the famly of all -element subsets of [n]. The famly s ( [n] s ( s 1 ( -regular. Thus we can nclude all of s [n] n F and ( n 1 ( n 1 at each vertex. could stll have some avalable degree ;.e. d = k 1 s 1 We add to F several more sets, of sze s + 1, whle makng sure that the degree of no pont grows above k 1,.e. no pont s contaned n more than d of these newly ncluded (s + 1-element sets. To acheve ths goal, we use a specal case of the celebrated theorem of Baranya [], due to Katona. Lemma 4.2 [4, Lemma, p. 179] Let a ( n r be an arbtrary nonnegatve nteger. Then there exsts an almost regular subhypergraph H H r n (that s the degrees of any two vertces n H dffer by at most one wth E(H = a edges. By choosng r = s + 1, a = d n s+1, Lemma 4.2 provdes us wth an almost regular (s + 1-unform hypergraph H wth a edges. deg H (v = D(H = a(s + 1 dn. v V As H s almost regular, no vertex has degree larger than d n H. Therefore F = ( s ( [n] H s a k-wse 0-Fscher famly wth the requred number of edges. Let us assume now that there exsts a k-wse 0-Fscher famly F wth F = F + 1. As F contans the frst F smallest subsets of [n], A F A s larger than A F A by at least max{ A : A F } s + 1, D(F D(F + s + 1. By our observaton D(F (k 1n, gvng us D(F (k 1n (s + 1. On the other hand by the defnton of F, D(F = d n ( dn (s (k 1 dn > s + 1 s (s (k 1 dn = (k 1n (s + 1, a contradcton. Hence we proved that F s a k-wse 0-Fscher famly of maxmum sze. The next theorem determnes m k l (n for most of the trples (n, k, l; n partcular when k s fxed and n s large enough. It reveals the fact that extremal k-wse l-fscher famles are not too exctng, 11

12 as they are usually obtaned from extremal k-wse 0-Fscher famles by the addton of l new ponts to each member. The proof s bascally dentcal to the one n Füred [7]; we nclude an adaptaton to our formulaton of the statement. Theorem 4. Let 2 n + 1 k 2, n l 1 be postve ntegers. Then m k l (n = max{mk 0 (n l, k 1} f k + l > n, m k l (n max{mk 0 (n l, n} f k + l n, Proof. Clearly m k l (n mk 0 (n l, snce from any k-wse 0-Fscher famly on n l ponts one can construct a k-wse l-fscher famly on n ponts just by addng the same l new ponts to each member. Consder a k-wse l-fscher famly F = {A 1,..., A m } of sze m = m k l (n. Case 1. There are k 1 sets n F whose ntersecton B s of sze l. Then every member of F must contan B. Thus the famly F \ B := {A \ B : A F} s a k-wse 0-Fscher famly on n l ponts, so F = F \ B m k 0 (n l. Hence m = mk 0 (n l. Case 2. The ntersecton of any k 1 members of F contans at least l + 1 ponts. Assume frst that k + l > n. The value of m cannot be less than k 1, snce any collecton of k 1 sets s a k-wse 0-Fscher famly. If m > k 1 then there exsts an l-element set B, whch s the ntersecton of k members A 1,... A k of F. There are at most k 1 ponts n [n] \ B, so there exsts an ndex j k, such that the ntersecton of the k 1 sets A 1,..., A j 1, A j+1,..., A k s also B. Ths contradcts the assumpton of Case 2 and proves m = k 1. Assume now that k + l n. Then clearly k 1 < n l + 1 = m 2 0 (n l mk 0 (n l m. Let A 1,..., A k 2 F be members such that the cardnalty of B = A 1... A k 2 s as small as possble. If there were > j > k 2 such that A B = A j B, that would mply l = A 1... A k 2 A A j = B A, contradctng our assumpton that there are no k 1 members of the famly havng l-element ntersecton. Thus A B determnes A unquely. The famly F B = {B : B = A B, > k 2}, defned on B ponts, s parwse l-fscher. Therefore the parwse Nonunform Fscher Inequalty mples that t could have at most B members. Snce F = m k, there exsts sets A k 1, A k F. Every one of the n l ponts n [n] k =1 A s not contaned n at least one of the A j s, 1 j k. A smple averagng argument then shows that the smallest (k 2-wse ntersecton B has cardnalty at most l + 2 k k 2 (n l = n k (n l. So F = F B + {A 1,..., A k 2 n k 2 (n l + (k 2 n (k 2 k ( k n l 1 n Now Theorems 1. and 1.4 are smple corollares of Lemma 4.1 and Theorem 4.. We only remark that m k 0 (n s monotone ncreasng n n and even mk 0 (n 1 domnates both k 1 and n n ther respectve range of nterest. 12

13 5 Remarks and Open Questons We beleve that Constructon A s optmal for every k provded n s large enough. The frst unknown case s that of the 4-wse oddtown famles, where the sze of the extremal famly s between n 12 and n 9. When a specfc L s gven, t looks plausble that one can mprove the constant multpler 1 of the logarthm n Proposton 1.5. It s not clear what the sharp bound n ths problem s, but probably the bound vares wth L. In the specal case L = {0} one can generalze Constructon A whch provdes a famly wth (k 1(n p log 2 (k 1 members, such that the cardnalty of any member s not dvsble by p but the cardnalty of the ntersecton of any k members s. We beleve that (as n the case p = 2 ths lower bound s tght. However, there are nontrval obstacles n modfyng the proof of Theorem 1.2 to show ths. A generalzaton of the constructon n Theorem 1. for arbtrary s = L s the followng. Let L = {0, 1,..., s 1}. Let G s,k be a maxmum packng of (s+1-element sets, such that each s-element set s contaned n at most k 2 members of G s,k. The trval combnatoral upper bound on G s,k s (k 2 ( n s /(s+1. For s = 2 and k =, the nfnte famly of Stener trple systems match ths upper bound. In general, the exstence of such tactcal confguratons s not known, but asymptotcally optmal famles could easly be constructed for every k and s = o(n by generalzng the constructon of [9, Theorem 8.1]. Let G α = { C ( [n] : } c α (mod n, s + 1 c C where α s an nteger. Then for any k 2 dstnct ntegers 0 α 1 < α 2 <... < α k 2 < n, =1 k 2 C α s a famly coverng each s-set at most (k 2-tmes. By averagng, there exsts a choce of α s for whch G = G s,k := =1 k 2 C α has at least (k 2 ( n s+1 /n members. Thus for s = o(n, G = ( k 2 s+1 + o(1( n s. Then n the famly F = s ( [n] =1 G every s-set s contaned n at most k 1 members,.e. all the k-wse ntersectons are n L. The sze of F s ( s+k 1 s+1 + o(1 ( n s. It s not hard to prove wth a method smlar to the one n Theorem 1. that for L = {0,..., s 1} F, ndeed, s asymptotcally optmal. We conjecture that t s actually optmal for any L, L = s. One queston we know very lttle about s the k-wse varant of the unform Ray-Chaudhur-Wlson Theorem. Suppose we are gven a set L of s ntegers. What s the sze of a maxmum famly F of l-element sets, such that the cardnalty of the ntersecton of any k members of F s n L. The upper bound s stll the usual (k 1 ( n s, whle we don t know any constructon havng more than (1 + o(1 ( n s elements. Acknowledgment We are grateful to N. Alon and B. Sudakov for helpful dscussons. Note added n proof. Our above conjecture about the optmalty of the sze ( s+k 1 s+1 + o(1 ( n s of the L-ntersectng famly F was settled recently by Füred and Sudakov [10]. 1

14 References [1] N. Alon, L. Baba, H. Suzuk, Multlnear polynomals and Frankl-Ray-Chaudhur-Wlson type ntersecton theorems, J. Combn. Theory Ser. A, 58 (1991, [2] L. Baba and P. Frankl, Lnear Algebra Methods n Combnatorcs, manuscrpt. [] Zs. Baranya, On the factorzaton of the complete unform hypergraph, Infnte and fnte sets (Colloq., Keszthely, 197; dedcated to P. Erdős on hs 60th brthday, North-Holland, Amsterdam, Colloq. Math. Soc. Janos Bolya, 10 (1975, Vol. I, [4] G. O. H. Katona, On Separatng Systems of a Fnte Set, Journal of Combnatoral Theory, 1 (1966, [5] E. R. Berlekamp, On subsets wth ntersectons of even cardnalty, Canad. Math.Bull. 12 (1969, [6] P. Frankl, R. M. Wlson, Intersecton theorems wth geometrc consequences, Combnatorca 1 (1981, [7] Z. Füred, On a problem of Deza and Frankl, Ars Combnatorca 1 (1982, [8] Z. Füred, On fnte set-systems whose every ntersecton s a kernel of a star, Dscrete Math. 47 (198, [9] Z. Füred, Matchngs and covers n hypergraphs, Graphs and Combnatorcs 4 (1988, [10] Z. Füred, B. Sudakov, Extremal set systems wth restrcted k-wse ntersectons, Journal of Combnatoral Theory (Seres A 105 ( [11] J. E. Graver, Boolean desgns and self-dual matrods, Ln. Alg. Appl. 10 (1975, [12] V. Grolmusz, Set-Systems wth Restrcted Multple Intersectons, The Electronc Journal of Combnatorcs 9 (2002, R8. [1] V. Grolmusz, B. Sudakov, On k-wse set-ntersectons and k-wse Hammng-dstances, J. Combnatoral Theory Ser. A 99 (2002, [14] K. N. Majumdar, On some theorems n combnatorcs relatng to ncomplete block desgns, Ann. Math. Stat. 24 (195, [15] V. T. Sós, Remarks on the connecton of graph theory, fnte geometry and block desgns, Theore Combnatore, Proc. Colloq., Rome, Vol. II, Att de Convegn Lnce 17 (197, Accad. Naz. Lnce, Rome. 14

15 [16] V. H. Vu, Extremal set systems wth upper bounded odd ntersectons, Graphs and Combnatorcs 1 (1997, no. 2, [17] V. H. Vu, Set systems wth weakly restrcted ntersectons, Combnatorca 19 (1999, no. 4,