Disjoint Systems. Abstract

Transcription

1 Disjoit Systems Noga Alo ad Bey Sudaov Departmet of Mathematics Raymod ad Beverly Sacler Faculty of Exact Scieces Tel Aviv Uiversity, Tel Aviv, Israel Abstract A disjoit system of type (,,, ) is a collectio C = {A 1,..., A m } of pairwise disjoit families of -subsets of a -elemet set satisfyig the followig coditio. For every ordered pair A i ad A j of distict members of C ad for every A A i there exists a B A j that does ot itersect A. Let D (,, ) deote the maximum possible cardiality of a disjoit system of type (,,, ). It is show that for every fixed, 1 lim D (,, ) = 1. This settles a problem of Ahlswede, Cai ad Zhag. Several related problems are cosidered as well. 1 Itroductio I Extremal Fiite Set Theory oe is usually iterested i determiig or estimatig the maximum or miimum possible cardiality of a family of subsets of a elemet set that satisfies certai properties. See [5], [7] ad [9] for a comprehesive study of problems of this type. I several recet papers (see [3], [1],[]), Ahlswede, Cai ad Zhag cosidered various extremal problems that study the maximum or miimum possible cardiality of a collectio of families of subsets of a -set, that satisfies certai properties. They observed that may of the classical extremal problems dealig with families of sets suggest umerous itriguig questios whe oe replaces the otio of a family of sets by the more complicated oe of a collectio of families of sets. Research supported i part by a Uited States Israel BSF Grat 1

2 I the preset ote we cosider several problems of this type that deal with disjoit systems. Let N = {1,,..., } be a elemet set, ad let C = {A 1,..., A m } be a collectio of pairwise disjoit families of -subsets of N. C is a disjoit system of type (,,, ) if for every ordered pair A i ad A j of distict members of C there exists a A A i which does ot itersect ay member of A j. Similarly, C is a disjoit system of type (,,, ) if for every ordered pair A i ad A j of distict members of C ad for every A A i there exists a B A j that does ot itersect A. Fially, C is a disjoit system of type (,,, ) if for every ordered pair A i ad A j of distict members of C there exists a A A i ad a B A j that does ot itersect A. Let D (,, ) deote the maximum possible cardiality of a disjoit system of type (,,, ). Let D (,, ) deote the maximum possible cardiality of a disjoit system of type (,,, ) ad let D (,, ) deote the maximum possible cardiality of a disjoit system of type (,,, ). Trivially, for every, D (,, 1) = D (,, 1) = D (,, 1) =. It is easy to see that every disjoit system of type (,,, ) is also a system of type (,,, ), ad every system of type (,,, ) is also of type (,,, ). Therefore, for every D (,, ) D (,, ) D (,, ). I this ote we determie the asymptotic behaviour of these three fuctios for every fixed, as teds to ifiity. Theorem 1.1 For every 1 lim D (,, ) = Theorem 1. For every Corollary 1.3 For every 1 lim D (,, ) = 1. 1 lim D (,, ) = 1. Theorem 1.1 settles a cojecture of Ahlswede, Cai ad Zhag [], who proved it for = [1]. The mai tool i its proof, preseted i Sectio, is a result of Fral ad Füredi [8]. The proof of Theorem 1., which settles aother questio raised i [] ad proved for = i [1], is more complicated ad combies combiatorial ad probabilistic argumets. This proof ad the simple derivatio of Corollary 1.3 from its assertio are preseted i Sectio 3.

3 Hypergraph decompositio ad disjoit systems I this sectio we prove Theorem 1.1. A -graph is a hypergraph i which every edge cotais precisely vertices. We eed the followig result of Fral ad Füredi. Lemma.1 ([8]) Let H = (U, F) be a fixed -graph with F = f edges. The oe ca place (1 o(1)) /f copies H 1 = (U 1, F 1 ), H = (U, F ),... of H ito a complete -graph o vertices such that U i U j for all i j, ad if U i U j = ad U i U j = B the B / F i, B / F j. Here the o(1) term teds to zero as teds to ifiity. Proof of Theorem 1.1 The lower boud for D (,, ) is a direct corollary of Lemma.1. Let (U, A) be the -graph cosistig of + 1 pairwise disjoit edges. By the lemma we ca place (1 o(1)) ( ) /( + 1) edge disjoit copies of this graph ito a complete -graph o vertices, so that ay two copies will have at most commo vertices. Therefore if we tae the edges of each copy as a family, we get (1 o(1)) ( ) /(+1) pairwise disjoit families which forms a disjoit system. Let H 1 = (U 1, A 1 ) ad H = (U, A ) be two such families. Sice U 1 U ad the family H 1 cosists of + 1 pairwise disjoit sets, there is a set A A 1 which does ot cotai ay poit of U 1 U. This A does ot itersect ay set of the family H. Therefore, our disjoit system is of type (,,, ). We ext establish a upper boud for D (,, ). Let C = {A 1, A,...} be a disjoit system of type (,, ). We deote by N 1, N 1 = 1, the set of all families of C cotaiig oe elemet, by N, N =, the set of those cotaiig from two up to elemets, ad by N 3, N 3 = 3, the set of those cotaiig more tha elemets. Sice sets i ay two oe-elemet families are disjoit we have 1 /. Let A i = {A 1,..., A t } be a family with t elemets. By the defiitio of a system of type (,,, ) we coclude that ay set B with the properties : caot be used as a elemet of ay other family. B = ; B t j=1a j ; B A j j (1) We ext boud the umber of such sets B from below. Choose ot ecessarily distict a j A j for j t such that a j / A 1. Put L = {a,..., a t }, the L 1. Let L be the family of all sets of the form L r = L Y r where Y r rages over all ( L ) elemet subsets of A 1. Clearly each such L r satisfies the properties (1). Moreover L r A 1 ad L. We claim that o -set ca satisfy the properties (1) for two or more families from C. Ideed assume this is false. Let B be a set which satisfies the properties (1) for two families A = {A 1,..., A l } 3

4 ad F = {F 1,..., F m }. By the defiitio of a disjoit system of type (,,, ) there exists a set A i A, 1 i l such that A i F j = for all j. Sice by (1) B m j=1 F j we coclude that B A i =, cotradictig (1) ad provig our claim. Therefore with each family A i i N we ca associate + 1 sets ( -sets of the form L r as above together with the set A 1 ) which caot be associated with ay other family ad are ot members of ay other family. I additio each family i N 3 cotais at least + 1 -sets. This implies that ( + 1) + ( + 1) 3. Therefore ( + 3 ) ( ) /( + 1). Together with the fact that 1 / we coclude that D (,,, ) + ( ) /( + 1) completig the proof. 3 Radom graphs ad disjoit systems I this sectio we prove Theorem 1.. We eed the followig two probabilistic lemmas. Lemma 3.1 (Cheroff, see e.g. [4], Appedix A) Let X be a radom variable with the biomial distributio B(, p). The for every a > 0 we have Let L be a graph-theoretic fuctio. P r( x p > a) < e a /. L satisfies the Lipschitz coditio if for ay two graphs H, H o the same set of vertices that differ oly i oe edge we have L(H) L(H ) 1. Let G(, p) deote, as usual, the radom graph o labeled vertices i which every pair, radomly ad idepedetly, is chose to be a edge with probability p. (See, e.g., [6].) Lemma 3. ([4], Chapter 7) Let L be a graph-theoretic fuctio satisfyig the Lipschitz coditio ad let µ = E[L(G)] be the expectatio of L(G), where G = G(, p). The for ay λ > 0 where m = ( ). P r( L(G) µ > λ m] < e λ / Proof of Theorem 1. Let 1 be the umber of families cotaiig oly oe elemet. The same argumet as i the proof of Theorem 1.1 shows that 1 /. This settles the required upper boud for D (,, ), sice all other families cotai at least sets. We prove the lower boud usig probabilistic argumets. We show that for ay ε > 0 there are at least 1 (1 ε)( ) families which form a disjoit system of type (,,, ), provided is sufficietly 4

5 large (as a fuctio of ε ad ). We first outlie the mai idea i the (probabilistic) costructio ad the describe the details. Let G = G(, p) be a radom graph, where p is a costat, to be specified later, which is very close to 1. We use this graph to build aother radom graph G 1, whose vertices are all -cliques i G. Two vertices of G 1 are adjacet if ad oly if the iduced subgraph o the correspodig -cliques i G is the uio of two vertex disjoit -cliques with o edges betwee them. Followig the stadard termiology i the study of radom graphs we say that a evet holds almost surely if the probability it holds teds to 1 as teds to ifiity. We will prove that almost surely the followig two evets happe. First, the umber of vertices i G 1 is greater tha (1 ε/) ( ). Secod, G1 is almost regular, i.e., for every (small) δ > 0 there exists a (large) umber d such that the degree d(x) of ay vertex x of G 1 satisfies (1 δ)d < d(x) < (1+δ)d, provided is sufficietly large. Suppose G 1 = (V, E) satisfies these properties. By Vizig s Theorem [10], the chromatic idex χ (G 1 ) of G 1 satisfies χ (G 1 ) (1 + δ)d + 1. Sice for ay x G 1 we have d(x) (1 δ)d, the umber of edges E of G 1 is at least (1 δ)d V. Hece there exists a matchig i G 1 which cotais at least (1 δ)d V /χ (G 1 ) (1 δ) V (1+δ) edges. This matchig covers almost all vertices of G 1, as δ is small, providig a system of pairs of -sets coverig almost all the ( ) -sets. Taig each pair as a family we have a disjoit system of size at least 1 (1 ε)( ) ad ε ca be made arbitrarily small for all sufficietly large. We ext show that the resultig system is a disjoit system of type (,,, ). Assume this is false ad let A = {A 1, A } ad B = {B 1, B } be two pairs where A 1 B i for i = 1,. Choose x 1 A 1 B 1 ad x A 1 B. Sice x 1 ad x belog to A 1 they are adjacet i G = G(, p). However, x 1 B 1, x B ad this cotradicts the fact that the subgraph of G iduced o B 1 B has o edges betwee B 1 ad B. Thus the system is ideed of type (,,, ) ad D (,, ) > 1 (1 ε) ( ) for every ε > 0, provided > 0 (, ε 1 ), as eeded. The proof that ideed G 1 has the required properties almost surely will be deduced from the followig two statemets. Fact 1. G = G(, p) satisfies the followig coditio almost surely. For every set X of vertices of G, the umber of vertices which do ot have ay eighbor i X is (1 + o(1))(1 p) ( ), where here the o(1) term teds to zero as teds to ifiity. 5

6 Fact. For ay c > 0, if is sufficetly large, G = G(, p) satisfies the followig coditio almost surely. For every set Y of 1 vertices of G, where c 1, the umber of -cliques of the iduced subgraph of G o Y is close to its expectatio, i.e., is ) (1 + o(1)) ( 1 where the o(1) term teds to zero as teds to ifiity. p ( ), The proof of Fact 1 is a stadard applicatio of Lemma 3.1 ad is thus left to the reader. Proof of Fact. Let H(Y, p) deote the iduced subgraph of G = G(, p) o a fixed set Y of vertices, where Y = 1. Let L be the graph-theoretic fuctio give by L(H ) = ( 1 1 )N(H ), where H is a graph o Y ad N(H ) deotes the umber of -cliques i H. The expected value of L(H(Y, p)) is easily see to be µ(l) = ( 1 ( 1 ) 1 ) p ( ), ad the expected value of N = N(H(Y, p)) is µ(n) = ( 1 ) p ( ). By the defiitio of L, if H1 ad H are two graphs o Y which differ oly i oe edge tha L(H 1 ) L(H ) 1, sice the umber of -cliques of H(Y, p) cotaiig a edge is at most ( 1 ). Thus, by Lemma 3. ) P r[ L(H(Y, p)) µ(l) > 3 4 ( 1 ] < e 3. Cosequetly, P r[ N(H(Y, p)) µ(n) > Sice ad p are costats, ad 1 c we coclude that = γ p ( ) = γµ(n), where γ = γ(, 1,, p) teds to 0 as teds to ifiity. ] < e 3. The total umber of possible sets Y is clearly less tha. Hece, the probability that for some Y, N(H(Y, p)) deviates by more tha γµ(n(h(y, p)) from its expectatio is less tha +1 e which teds to zero as teds to ifiity. This completes the proof of Fact. Returig to the proof of the theorem cosider a -clique X i G. The degree d of X as a vertex of G 1 is the umber of -cliques i the iduced subgraph of G o the set of all vertices which have o eighbors i X. By Facts 1 ad each such degree is almost surely 1 (1 + o(1)) p ( ) 6 3,

7 where 1 = (1 + o(1))(1 p) ( ). Therefore, almost surely G 1 is almost regular ad the degrees of its vertices ted to ifiity with. I a similar maer, Fact applied to the set Y of all vertices of G implies that the umber of -cliques i G = G(, p) (which is the umber of vertices of G 1 ) is almost surely Fix p < 1 so that p ( ) > 1 ε 4 i G 1 is more tha as eeded. (1 + o(1)) p ( ). for the required ε. With this p, almost surely the umber of vertices (1 ε/), Therefore, our procedure produces, with high probability, a disjoit system of the required type with at least 1 (1 ε)( ) pairs, completig the proof. Remar. By combiig our method here with the techique of [8] we ca prove the followig extesio of Lemma.1, which may be useful i further applicatios. Sice the proof is similar to the last oe, we omit the details. Propositio 3.3 Let H = (U, F) be a fixed -graph with F = f edges ad let g deote the maximum cardiality of a itersectio of two distict edges of H. The oe ca place (1 o(1)) /f pairwise edge-disjoit copies H 1 = (U 1, F 1 ), H = (U, F ),... of H ito a complete -graph o vertices such that U i U j for all i j, ad such that if for some i j there is a F j F j so that F j U i g + the there is a F i F i so that F j U i F i. Here the o(1) term teds to zero as teds to ifiity. Proof of Corollary 1.3 Let 1 be the umber of oe elemet families i a disjoit system of type (,,, ). The trivial argumet used i the proofs of Theorems 1.1 ad 1. shows that 1 / ad thus implies that D (,, ) + 1. As observed i Sectio 1, D (,, ) D (,, ) ad hece, by Theorem 1., the desired result follows. Acowledgmet We would lie to tha the referees for several helpful commets ad suggestios. 7

8 Refereces [1] R. Ahlswede, N. Cai ad Z. Zhag, A ew directio i extremal theory for graphs, to appear i JCISS. [] R. Ahlswede, N. Cai ad Z. Zhag, Higher level extremal problems, to appear. [3] R. Ahlswede ad Z. Zhag, O cloud-atichais ad related cofiguratios, Discrete Math. 85 (1990), [4] N. Alo ad J. H. Specer, The Probabilistic Method, Wiley, [5] I. Aderso, Combiatorics of Fiite Sets, Claredo Press, [6] B. Bollobás, Radom Graphs, Academic Press, [7] B. Bollobás, Combiatorics, Cambridge Uiversity Press, [8] P. Fral ad Z. Füredi, Colored pacig of sets i combiatorial desgi theory, Aals of Discrete Math. 34 (1987), [9] Z. Füredi, Matchigs ad covers i hypergraphs, Graphs ad Combiatorics 4 (1988), [10] V. G. Vizig, Colorig the vertices of a graph i prescribed colors (i Russia), Disret. Aaliz. No. 9, Metody Disret. Aal. v. Teorii Kodov i Shem 101 (1976),