Almost intersecting families of sets

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1 Almost itersectig families of sets Dáiel Gerber a Natha Lemos b Cory Palmer a Balázs Patkós a, Vajk Szécsi b a Hugaria Academy of Scieces, Alfréd Réyi Istitute of Mathematics, P.O.B. 17, Budapest H-1364, Hugary b Cetral Europea Uiversity, Departmet of Mathematics ad its Applicatios, Nádor u. 9, Budapest H-1051, Hugary April 7, 010 Abstract Let us write D F (G = {F F : F G = } for a set G ad a family F. The a family F of sets is said to be ( l-almost itersectig (l-almost itersectig if for ay F F we have D F (F l ( D F (F = l. I this paper we ivestigate the problem of fidig the maximum size of a ( l- almost itersectig (l-almost itersectig family F. AMS Subject Classificatio: 05D05 Keywords: extremal set theory, itersectio theorems, Sperer-type theorems 1 Itroductio We will use stadard otatio: [] deotes the set of the first positive itegers {1,,...,} ad for ay set X we write ( X k for the family of all k elemet subsets of X ad X for the power set of X. For a family F X we write F = {F : F F}. We will say that a family F is itersectig if F,G F implies F G. Moreover, F is trivially itersectig if F F F ad a family F is called Sperer if there exist o F,G F with F G. Oe of the basic results about extremal set families is due to Erdős, Ko ad Rado [7] ad states that if k ad F ( [] k is itersectig, the the size of F is at Research supported by Hugaria Natioal Scietific Fud, grat umber: OTKA K-6906 correspodig author, patkos@reyi.hu 1

2 most 1 k 1, furthermore if k <, the equality holds if ad oly if F is a trivially itersectig family. The o-uiform versio (i.e. whe sets i F eed ot to have equal size of the above theorem is also due to Erdős, Ko ad Rado. However, it is rather a easy exercise to prove that ay itersectig family G [] ca be exteded to a itersectig family G of size 1 ad there exists o itersectig family of larger size. These theorems attracted the attetio of may researchers. Several geeralizatios have bee proved, the itersectig coditio has bee relaxed or stregtheed i may ways. Oe relaxatio is to allow some fixed umber of disjoit pairs formed by members of the family F [1, 5, 8]. I this paper we cosider families F where for ay F F there are at most a fixed umber of sets disjoit from F. More precisely, for ay set G ad family F let D F (G = {F F : F G = } be the subfamily of the sets disjoit from G. We say that a family F is ( l-almost itersectig if for every set F F we have D F (F l, ad we say that a family F is l-almost itersectig if for every set F F we have D F (F = l. We will address the problem of fidig the largest size of a ( l-almost itersectig (l-almost itersectig family F [] both i the uiform ad i the o-uiform case. Clearly, l = 0 gives back the origial problem of Erdős, Ko ad Rado. The rest of the paper is orgaized as follows: i Sectio we cosider k-uiform l-almost itersectig families. Amog others we prove the followig cojecture for k =. Cojecture 1.1. For ay k there exists l 0 = l 0 (k such that if l l 0 ad F is a k-uiform l-almost itersectig family, the F (l + 1 ( k k 1. The followig costructio shows that if true, Cojecture 1.1 is sharp: {F {i} : F ( [k ] k 1,i {k 1, k,..., k + l 1}}. I Sectio 3 we cosider o-uiform l-almost itersectig families ad solve the problem completely if l is 1 or. I Sectio 4 we prove that for ay fixed k ad l if 0 (k,l, the the largest k- uiform ( l-almost itersectig family is a trivially itersectig family. Determiig the smallest possible 0 remais ope except for the case l = 1 for which we prove the miimum of 0 (k, 1 is k +. Sectio 5 deals with o-uiform ( l-almost itersectig families. We settle the problem for l = 1,. For larger l we cojecture the followig. Cojecture 1.. For ay positive iteger l there exists 0 = 0 (l such that if 0 ad F [] is a ( l-almost itersectig family, the { i=/ F i if is eve 1 / + i= / i if is odd,

3 ad equality holds if ad oly if F is the family of sets of size at least / ad (if is odd the sets of size / ot cotaiig a fixed elemet of []. Restrictive case - Uiform families I this sectio we ivestigate k-uiform l-almost itersectig families. The followig otio ad Theorem.1 will play a very importat role i our proofs. The collectio of pairs of sets (A i,b i m i=1 are said to form a cross-itersectig family if for ay 1 i,j m we have A i B j = if ad oly if i = j. The mai theorem about cross-itersectig families of pairs is the followig result. Theorem.1 (Bollobás [3]. If the pairs (A i,b i m i=1 form a cross-itersectig family, the the followig iequality holds: m i=1 1 ( Ai + B i A i 1, i particular if A i k ad B i l for all 1 i m, the m ( k+l k ad equality holds if ad oly if the pairs are all possible partitios ito sets of size k ad l of some (k + l-set X. The followig easy corollary settles Cojecture 1.1 whe l = 1. Corollary.. If F is a k-uiform 1-almost itersectig family, the F ( k k ad equality holds if ad oly if F = ( X k with X = k. Proof. For ay k-uiform 1-almost itersectig family F = {F 1,F,...,F m }, let A i = F i ad let B i deote the oly set i F which is disjoit from F i. The the pairs (A i,b i m i=1 form a cross-itersectig family ad we are doe by Theorem.1. The ext lemma shows that for ay positive itegers k ad l there exists a upper boud o the size of a k-uiform l-almost itersectig family which is idepedet of the size of the groud set. Lemma.3. For ay k,l N where l > 0, if F is a k-uiform l-almost itersectig family, the we have F l ( kl kl. Proof. Let F = {F 1,F,...,F m } ad let I j = {i [m] : D F (F j = D F (F i }. Note that for ay j m we have I j l as otherwise ay G D F (F j would be disjoit from at least l + 1 sets of F, a cotradictio. Let J [m] be a set of idices so that I j I j for ay j,j J ad J m l. For every j J we defie the sets A j := i Ij F i, B j := G DF (F j G. Clearly they form a cross-itersectig family ad A j, B j kl, hece we are doe by Theorem.1. 3

4 We fiish this sectio by settlig the case k = of Cojecture 1.1. Istead of talkig about -uiform families we state our result i the laguage of graphs. Theorem.4. If G is a l-almost itersectig graph with o isolated vertices, the G has at most l+ edges ad the uique extremal graph is K,l+1 provided l 1, 3, 5, 6. For l = 1, 3, 6 the uique extremal graphs are K 4,K 5,K 6 respectively, ad for l = 5 there are two extremal graphs: K,6 ad the complemet of a matchig o 6 vertices. Proof. If G is ot coected, the there is a cut C 1,C of V (G with e(c 1,C = 0, e(c 1,e(C > 0. But the by the l-almost itersectig property of G we obtai e(c 1,e(C l ad thus e(g l. Hece we ca suppose G is coected. Let e be the umber of edges ad the umber of vertices. If there are exactly l edges disjoit from ay fixed edge, the for ay edge (u,v we have d(u + d(v 1 = e l. Thus for ay vertex all its eighbors have fixed degree. We distiguish two cases. Case 1. G is d-regular for a iteger d. The clearly e = d + l 1 ad d = e = (d + l 1, hece ( 4d = l. Sice d is the degree of all vertices i G, it is at most 1. Hece l = ( 4d (d 3d = (d 3 9 ad thus d 3 l + l 1. The e = d+l 1 l+ + 1+, which is strictly less tha l + if l > 8. If d = 1, the G = K ad every edge is disjoit from ( other edges. As = l 8, thus we obtai 7 ad K4,K 5 ad K 6 by the above, we must have do cotai more edges tha K,,K,4 ad K,7 respectively. Otherwise d ca be supposed, ad by repeatig the previous calculatio we get e l + l 1 + 1, which is strictly less tha l + if l > 5. For l = 5 there is equality here, this gives the two extremal graphs. l = d(d is impossible for l < 5, hece we ca suppose d 3. The simple calculatio used i the previous paragraphs gives e l + l 7 i this 4 case, which is always strictly less tha l +. Case. G is ot regular ad thus the degrees give a -colorig of G. Let us deote the two color classes by A ad B, their cardiality by a ad b, ad the degrees by d A ad d B. The e = ad A = bd B = 1(ad A + bd B = l + d A + d B 1. Clearly d A b ad d B a, hece we get ab a b l 1 ad e ab. Let a b (i fact a = b happes oly if G is regular, hece we ca assume a < b. The a = 1 meas all the edges itersect ad a = gives at most (l +1 edges, with equality oly i the case of K,l+1. Hece we ca suppose a 3. By a + b = we obtai ab 3( 3, thus 9 ab a b l 1 ad l + 5. If l > 4 the it implies e ab < l +. If l 4 the the iequalities a + b 7 ad b > a 3 give a = 3 ad b = 4, but the d A = 4 ad d B = 3 is ecessary by the biregularity of G. Thus we must have G = K 3,4, hece l = 6, a cotradictio. 4

5 3 Restrictive case - No-uiform families I this sectio we cosider l-almost itersectig families that are ot ecessarily uiform. We start our ivestigatios with a useful defiitio ad a propositio valid for arbitrary l. For ay family F of sets, the comparability graph G(F is the graph with vertex set V (G = F ad edge set E(G = {(F 1,F : (F 1 F (F F 1 }. Propositio 3.1. If F is a l-almost itersectig family, the all coected compoets of G(F have size at most l. Proof. For ay pair F 1,F F of sets with F 1 F, we have D F (F 1 = D F (F as all sets disjoit from F are disjoit from F 1 as well ad D F (F = l for ay F F. We obtai that if F,F lie i the same compoet of G(F, the we have D F (F = D F (F. Therefore if a compoet C of G(F cosisted of more tha l vertices, the D F (H > l would hold for ay set H D F (F with F C. As a special case of Propositio 3.1 we obtai that a l-almost itersectig family does ot cotai a l-fork (a family of l + 1 sets F 0,F 1,...,F l with F 0 F i for all 1 i l, hece the followig theorem of De Bois ad Katoa ca be used (a weaker versio was obtaied earlier by Thah [13]. Theorem 3. (De Bois, Katoa [6]. If a family F [] does ot cotai a r-fork, the F (1 + r + O( 1 /. Corollary 3.3. If F is a l-almost itersectig family, the F (1+ l +O( 1 /. As we will see, this boud is asymptotically tight if l = 1. If l = the boud is off by a factor of as show by Theorem 3.14 ad we cojecture that it is eve further from the truth for larger values of l, but this is the best boud we have at the momet. Now let us cosider the case l = 1. Theorem 3.4. If F [] is a 1-almost itersectig family, the { if is eve / F ( 1 if is odd, / 1 ad equality holds if ad oly if F = ( ( [] / provided is eve ad F = {F [] / : x F } {F ( [] / : x / F } for some fixed x [] provided is odd. 5

6 Proof. Let F be a 1-almost itersectig family of maximum size. By Propositio 3.1 we kow that F is Sperer ad thus if is eve, we are doe by Sperer s theorem [1]. Suppose is odd. Clearly, F cosists of disjoit pairs. Let us cosider the subfamily F that cosists of the smaller set from each pair ad let F = F \ F. The F is Sperer, itersectig ad F for all F F. We use a theorem of Bollobás [4] that deals with families of this type. We oly state the result for odd. Theorem 3.5 (Bollobás [4]. If is odd ad F [] is a itersectig Sperer family with the property that F holds for all F F, the ( 1 F, / 1 ad equality holds if ad oly if F = {F ( [] / : x F } for some fixed x []. As F = F it follows that F ( 1 / 1 ad F = {F ( [] / : x F } for some fixed x []. If F = F = {F : F F }, the we are doe. Otherwise there exists F F with F /. The the family F = (F \ {F } {F } (where F is the uique set i F which is disjoit from F satisfies the coditios of Theorem 3.5 ad thus F = {F ( [] / : y F } for some fixed y [] but both x = y ad x y is impossible. Let us cotiue with the case l = by defiig -almost itersectig families. Costructio 3.6. If = k +, the the followig -almost itersectig family has size ( k k = ( 1 + o(1( / : let G1, G be a partitio of ( [k] k such that G G1 if ad oly if G G. The G = G 1 {G {k +1} : G G 1 } G {G {k +} : G G } possesses the required properties. Similarly, if = k + 1, the the followig family is -almost itersectig: let G 1 = {G ( [k 1] k 1 : x G}, G = {G ( [k 1] k : x / G} for some fixed x [] ad defie G = G 1 {G {k} : G G 1 } G {G {k +1} : G G }. The G possesses the required property ad has size 4 ( k k = ( 1 + o(1 /. I the remaider of this sectio we show that these costructios are best possible. Let F [] be a -almost itersectig family ad let us write F = F 1 F U F L where F U = {F F : F F,F F }, F L = {F F : F F,F F } ad F 1 = F \ (F U F L. Propositio 3.7. If F F 1, the for ay G F we have G / F. 6

7 Proof. If such a G was i F, the the two sets i F disjoit from G would be subsets of F, thus at least oe of them should be a proper subset of F. That would cotradict F F 1. Propositio 3.8. F L F U =. Proof. The compoet of a set F F L F U i the comparability graph G(F would have size at least 3 cotradictig Propositio 3.1. Propositio 3.9. For ay F F L (F F U there exists exactly 1 set F F U (F F L with F F (F F. Proof. This is the l = special case of Propositio 3.1. Corollary If F F L, the for ay G F we have G / F. Proof. Ay such G would cotai the two sets i F that are disjoit from F with F F. Propositio For ay -almost itersectig family F [] there exists aother such family G with F = G such that (i G G U implies G G U, (ii for ay G 1,G G with G 1 G we have G \ G 1 = 1. Proof. If F U F L = the G = F satisfies (i ad (ii. Otherwise let F 1,F F with F 1 F. The there exist distict sets F,F F which are disjoit from F ad therefore from F 1 ad all other sets i F meet both F 1,F. Thus replacig F 1 with ay set G satisfyig F 1 G F, F \ G = 1 will ot violate the -almost itersectig property of the ew family (as G is disjoit from F ad F ad as a superset of F 1 meets all other sets of the family. By repeatig this operatio we ca obtai a family satisfyig (ii. Suppose that i the above situatio we have F,F F. The either of these sets ca be replaced with F without violatig the the -almost itersectig property of the ew family. By repeatig this operatio we ca obtai a family satisfyig (i. We will call a -almost itersectig family good if it satisfies (i ad (ii of the above propositio. If G is good, the for ay G G U there exist G 1,G G such that G 1 G,G G, G \ G 1 = G \ G = 1. Let x be the oly elemet of G \ G 1 ad y be the oly elemet of G \ G ad let G = G \ {x} {y} = G 1 {y}. Propositio 3.1. Let G [] be a good -almost itersectig family. The for ay G G U the sets G 1,G,G defied as above satisfy the followig: 7

8 (i G / G, (ii G / G, (iii the oly set i G G cotaied i G is G 1, (iv the oly set i G G cotaiig G is G, (v {G : G G U } is a Sperer family. Proof. The statemets (i,(ii follow from Propositio 3.9 used for G 1 ad G. A set G G cotradictig (iii would be a third set i G which is disjoit from G, while for a set G G cotradictig (iii its complemet G would cotradict Propositio 3.9 for G. The statemet of (iv follows similarly usig G 1 i place of G. Fially, (v follows from the fact that G G would imply that G ad G 1 are disjoit ad therefore there would be three sets i G which are disjoit from G. Lemma Let G [] be a good -almost itersectig family. The the followig iequality holds: G G L G G U G G G ( G ( G G 1 G ( ( G ( 1 + ( G ( G 1 ( G G ( 1 ( G ( + G 1 G ( G 1 G ( G ( G 1. (1 Proof. First ote that / G as there would be oly oe other set i G, which could ot be disjoit from two sets. Let us cosider the pairs (G, C with G G, C is a maximal chai i [] ad G C. For ay set G there are G!( G! chais cotaiig G. Thus the umber of pairs is exactly G G G!( G!. O the other had by Propositio 3.8 every chai C may cotai at most sets from G, thus the umber of such pairs is! + c c 0 where c i is the umber of chais cotaiig i sets from G. By Propositio 3.9 we have c = G G G!( G 1!. L We would like to get a lower boud o c 0. Let us cosider the set S G of chais that cotai the complemet G of a fixed set G G 1 G L. By Propositio 3.7 ad Corollary 3.10, if G G 1 G L, there ca be o G G with G G. A k-subset of G is cotaied i G!k!( G k! chais that go through G. Thus, if G G 1, as the empty set caot be i G ad there are exactly sets i G which are disjoit from G, we obtai that there are at least ( G ( G 1! G! chais i S G that do ot cotai ay set from G. If G G L, the we kow that the sets i G cotaied i G have size G 1 ad G ad thus the umber of chais 8

9 i S G that avoid G is G!(( G! ( G 1! ( G!. By defiitio there do ot exist sets G,G i G 1 G L with G G, thus we have S G S G = for ay distict G,G G 1 G L. Fially, let us cosider sets G G U. By Propositio 3.1 (ad usig the defiitios precedig the propositio we get that ay chai that cotais G but cotais either G 1 or G avoids G G. Therefore these chais are differet from all chais i G G1 G LS G ad cotai o sets from G. The umber of such chais for oe fixed G G U is ( G! ( G 1!(( G! ( G 1!. Note that by Propositio 3.1 (v we have S G S G = for ay G,G G U. Addig the above observatios together we obtai G!( G!! + G!( G 1! G G G G L G G 1 ( G G!( G 1! G! (( G! ( G 1! ( G! G G L ( G! ( G 1!(( G! ( G 1!. ( G G U Rearragig ad dividig by! yield the statemet of the lemma. Clearly, the mai term of the LHS of (1 is the first term G G ad all other ( G terms are egligible compared to this, thus Costructio 3.6 is asymptotically best possible. We eed a little more work to prove the exact boud. Theorem Let F [] be a -almost itersectig family, the { ( if is eve F 4 ( 3 if is odd, / ad this boud is best possible as show by Costructio 3.6. Proof. Without loss of geerality we may assume that F is good. The we kow that sets i F U F L come i pairs with sizes differig by 1. Let us cosider the summads i (1. Let a k deote the sum of all summads from sets i F 1 of size k ad let b k deote the sum of all summads from pairs i F U F L such that the smaller set i the pair has size k. Clearly, if m = mi{a k, b k : 1 k 1}, the F 1 m. For coveiece we rather work with a k =!a k = k!( k! k!( k 1! 9

10 ad b k =!b k = k!( k! + (k + 1!( k 1! k!( k 1! k!( k! (k + 1!( k! k!( k 1! + k!( k! First we claim that if is eve, the miimum of a k over k is a / it is a / = a /. Ideed, let ad if is odd, A k = a k+1 a k = k!( k![(k + 1( k ( k 1 ] ad observe that the expressio i the brackets is quadratic i k. Furthermore if is odd, oe of its roots is betwee 3 ad ad the other is /. If is eve, it has a root betwee / 1 ad /. This proves our claim. Next we claim that the same holds for each b k, i.e. if is eve, the miimum of b k over k is b / ad if is odd, it is b / = b /. This ca be show by a similar (but a bit more tedious calculatio ivolvig B k = b k+1 b k = k!( k 3! [(k + 1(k + ( k ( k( k 1( k (k + 1(3 k 4 + (3 k ( k ]. Fially, by substitutig, a / > 1 b / ad the theorem follows as the size of the families i Costructio 3.6 is exactly b /. The case whe l > remais usolved. We fiish this sectio with a questio which is related to the l = case. What is the maximum size of a family that satisfies the Sperer property i additio to beig -almost itersectig (or l-itersectig for some l? Is the followig costructio optimal or asymptotically optimal? Costructio Let F be a optimal 1-almost itersectig family o [ l 1] as i Theorem 3.4. The G = F ( [ l,] 1 = {F {x} : F F,x [ l,]} is a Sperer ad l-almost itersectig family of size ( l+1 + o(1 /. l+1 4 The less restrictive case - Uiform families I this sectio we cosider k-uiform ( l-almost itersectig families. First, we prove that if is large eough, the this relaxatio of the itersectig property does ot allow us to obtai a larger family, tha what the Erdős-Ko-Rado result states. Propositio 4.1. For ay k,l N there exists 0 = 0 (k,l such that if 0 ad F ( ( [] k is a ( l-almost itersectig family, the F 1 k 1 with equality if ad oly if F is the family of all k-sets cotaiig a fixed elemet of []. 10

11 Proof. If F is itersectig, the we are doe by the Erdős-Ko-Rado theorem. If F 1,F F are disjoit, the ay F F \ (D F (F 1 D F (F should meet both F 1 ad F ad thus F k ( ( k + l which is smaller tha 1 k 1 if is large eough. The argumet of Propositio 4.1 gives 0 (k,l O(k 3 + kl. Usig a theorem of Hilto ad Miler [9] that states that a o-trivially itersectig family has size at most ( ( 1 k 1 k 1 k oe ca obtai a boud 0 (k,l = O(k l which is better tha the previous boud if l = o(k. Fidig the smallest possible 0 (k,l seems to be a iterestig problem. Obviously, if l ( ( m k, the [k+m] k is ( l-almost itersectig, furthermore if l ( ( k 1 k 1, the the trivially itersectig family {F [] k : 1 F } is ot maximal. The followig theorem states that k + is a good choice for 0 (k, 1 provided k 3. Note that the case k = is trivial as if there exist F 1,F F with F 1 F =, the all other sets i F must itersect both F 1 ad F ad thus be subsets of F 1 F ad therefore F 6. Comparig this to the size of the trivially itersectig family gives 0 (, 1 = 7. Theorem 4.. If k 3 ad k+ ad F ( [] k is a ( 1-almost itersectig family, the F 1 k 1 with equality if ad oly if F is the family of all k-sets cotaiig a fixed elemet of []. Proof. We will use Katoa s cycle method [10]. We call a subset S of [] a iterval i a cyclic permutatio π of [] if S = {π(i,π(i+1,...,π(i+ S 1} for some i ad additio is modulo. We say that two sets S,T are separated i a cyclic permutatio π if there are disjoit itervals S,T i π with S S,T T ad S T = []. Let F 0 = {F F : D F (F = 0}, F 1 = {F F : D F (F = 1}. We defie two types of objects ad give them weights: 1. for (F,π with F F 0 ad F a iterval i π, give the weight 1 k ;. for ({F,F },π with F,F F 1 ad F,F separated i π, give the weight. Lemma 4.3. If k ad F ( [] k is a ( 1-almost itersectig family, the for ay cyclic permutatio π of [] the sum of the weights of all objects havig π as secod coordiate is at most 1. Proof. For simplicity we idetify objects with their first coordiates. We say that for ay cyclic permutatio π there is a milestoe betwee ay cosecutive elemets π(i,π(i + 1. For ay object with secod coordiate π we defie two separatig milestoes: if the object is of type 1, the the milestoes lyig betwee the iterval ad its complemet are the separatig milestoes. If the object is of type, the 11

12 there exist itervals S,T showig this fact ad we take the milestoes betwee S ad T. We claim that ay milestoe ca belog to at most oe object. Otherwise if both objects are of type 1, the to be differet they must lie o the two sides of the milestoe ad as k they do ot itersect. If oe of the objects F is of type 1 ad the other {S,T } is of type, the F S or F T ad thus either F T = or F S =. Fially, if both objects {S 1,T 1 }, {S,T } are of type, the S 1 S or S S 1 ad thus S 1 T = or S T 1 =. As each object has separatig milestoes, the lemma immediately follows if all objects are of type. Let us assume that there is at least oe object F of type 1 ad we say that F crosses the milestoes betwee ay two of its elemets. The for ay other object O, the iterval F crosses at least oe of the separatig milestoes of O. This is clear if O is of type 1 while if O = {S,T }, the ot crossig ay of the milestoes would mea F S or F T ad thus F T = or F S =. But clearly a iterval of size k may cross at most k 1 itervals, thus we obtai that i this case there are at most k objects with secod coordiate π ad as 1 k the lemma follows. Let α deote the umber of cyclic permutatio i which the disjoit k-sets S ad T are separated. Clearly, α does ot deped o the actual choice of S ad T. With this otatio ad Lemma 4.3 we obtai the followig iequality F 0 k!( k! 1 k + F 1 α 1 ( 1! Thus we will be doe if we ca prove that k!( k! 1 k < α1. (3 We calculate α i the followig way: cosider the family G = ( [k] k ad cout the objects ({G 1,G },π with G 1,G G ad separated i π. O the oe had, this is ( 1 k k α. O the other had, this is k( 1! as ay cyclic permutatio π of [] cotais exactly k separated pairs from G. Thus we obtai α = k( 1! ( k k. After substitutig this ito (3 we eed oly to verify k!( k! 1 k k( 1! < ( k k 1

13 which is equivalet to (k! ( 1! < k!k ( k!. This holds if k + provided k 5. For k < 5 the above iequality holds if k+3. The cases k = 3, = 8 ad k = 4, = 10 ca be verified by chagig the weight of all objects ({F,F },π to 1/k whe at least oe of F ad F is a iterval i π. Oe ca easily check that Lemma 4.3 holds with the modified weights. We leave the details to the reader. Note that sice the family ( ( [k] k is 1-almost itersectig ad has size k k > ( k+1 1 k 1, 0 (k, 1 = k + is best possible. We fiish this sectio with a easy double coutig proof that settles the case of = k + 1. Propositio 4.4. If F ( [k+1] k is a ( 1-almost itersectig family, the F ad equality holds if ad oly if F is the family all k-sets ot cotaiig a fixed ( k k elemet of [k + 1]. Proof. Let us double cout the pairs (F,G with F F, G / F. O oe had this is at least k F as for ay F F out of the k + 1 may k-sets disjoit from F at most 1 ca be i F. O the other had the umber of such pairs is at most ( ( ( k+1 k F (k + 1. We obtai ( k+1 k F (k + 1 k F ad by rearragig we get the stated boud o F. To characterize the case of equality ote that all lower ad upper bouds i the previous argumet hold with equality if ad oly if F is a 1-almost itersectig family. Thus we are doe by Corollary.. 5 The less restrictive case - No-uiform families I this sectio we cosider the problem of fidig the maximum size of a ( l- almost itersectig family F []. Theorem 5.3 will settle the case of l = 1, we prove Cojecture 1. for the case l =. Remark 5.1. A family where all supersets of ay member of the family belog to the family as well is called a upset. For ay ( l-almost itersectig family F there is aother family F of the same size which is a upset. Ideed, if for two sets F,G we have F G, F F, G / F, the F \ {F } {G} is ( l-almost itersectig provided F is as well. Thus it is eough to prove the upper boud i Cojecture 1. for upsets, furthermore uiqueess for upsets implies uiqueess for arbitrary families. Ideed, cosider a ( l-almost itersectig family F of maximum size. By applyig repeatedly the above operatio, we obtai a family F such that 13

14 F = F ad F \ {F } {G} = F, where F G,F F,G / F ad F is the uique family described i Cojecture 1.. but the D F (F 1 ( 1 which cotradicts the ( l-almost itersectig property of F. We start with a lemma that we will use i provig Theorem 5.3 ad Theorem 5.5 ad might be useful i verifyig Cojecture 1.. Lemma 5.. Let l be a positive iteger. If a ( l-almost itersectig family F is a upset such that the size m of a miimum set i F is at most l, the there exists aother ( l-almost itersectig family F with F F such that the size of a miimum set i F is m + 1. Furthermore, if m < l, the F < F. Proof. Let us write F i = {F F : F = i} ad defie the bipartite graph G(V 1, V,E where V 1 = F m, V = {G ( [] m 1 \ F m 1 : F F m,f G = }, E = {(F,G : F G = }. Clearly, for ay G V we have d(g m + 1. Also, for ay F F m = V 1 we have d(f m l + 1 as there are m sets of size m 1 which are disjoit from F ad at most l 1 of them belog to F by the ( l-almost itersectig property sice F beig a upset guaratees that F F provided D F (F. It follows that V 1 V, furthermore if m < l, the V 1 < V. Let us defie F = (F \V 1 V. By the above, we have F F ad if m < l, the F < F. We still have to show that F is ( l-almost itersectig. Sice all sets i V have size m 1 ad the miimum sets i F have size m + 1 for ay set G V we have D F (G = 1. Also, for ay set F F F if F > m + 1, the D F (F D F (F ad thus D F (F D F (F l. Fially, let us cosider a set F F F with F = m + 1. If there was o set F F i F m, the D F (F = D F (F, while if there was, the the oly set i D F (F \ D F (F is F, therefore D F (F D F (F + 1. To fiish the proof of the lemma observe that as F F we have D F (F < D F (F l. The ext theorem cosiders the case l = 1. Theorem 5.3. If ad F [] is a ( 1-almost itersectig family, the { i=/ F i if is eve 1 / + i= / i if is odd, ad equality holds if ad oly if F is the family of sets of size at least / ad (if is odd the sets of size / cotaiig a fixed elemet of []. Proof. Let F be a ( 1-almost itersectig family of maximum size. Remark 5.1 shows that we may assume that F is a upset ad by Lemma 5. we may assume 14

15 that the size of a miimum set i F is at least 1. If is eve, this meas F {F [] : F /} ad we are doe. If = k + 1 is odd, the Lemma 5. gives F {F [k + 1] : F k}. We claim that ( ( [k+1] k+1 F. Ideed, if G [k+1] k+1, the DF (G 1, thus the oly reaso for which G could ot be i F is G F ad there exists G F such that G G. But the G would belog to F as F is a upset, a cotradictio. As ( [k+1] k+1 F, we kow that for every F Fk we have F F ad thus F k = {F F : F = k} must form a itersectig family. Thus we are doe by the Erdős-Ko-Rado Theorem. Theorem 5.3 ca also be derived from a result by Beráth ad Gerber []. We defie a family F to be (p,q-chai itersectig if A 1 A... A p, B 1 B... B q with A i,b j F implies A p B q. Theorem 5.4 (Beráth, Gerber []. If F [] is (p,q-chai itersectig, the { i=( p q+3/ F i if p q is odd ( 1 ( p q+3/ 1 + i= ( p q+3/ i if p q is eve. Theorem 5.3 follows from Theorem 5.4 by lettig p = 1, q = as the (1,-chai itersectig property is equivalet to the coditio that D F (F is Sperer for all F F. Beráth ad Gerber also deal with the case of equality. We do ot state their complete result for sake of brevity. The ext theorem states that Cojecture 1. is true if l =. Theorem 5.5. If ad F [] is a ( -almost itersectig family, the { i=/ F i if is eve 1 / + i= / i if is odd, ad equality holds if ad oly if F is the family of sets of size at least / ad (if is odd the sets of size / ot cotaiig a fixed elemet of []. Proof. Let us cosider two cases accordig to the parity of. If = k + 1 is odd, the by Remark 5.1 ad Lemma 5. we ca assume that F is a upset ad all sets i F have size at least k. Therefore all sets F of size at least k + belog to F as D F (F =. We claim that if F is maximal, the ( [k+1] k+1 F. Ideed, if G = k +1, the D F (G 1, thus the oly reaso for which G could ot be i F is that G F ad D F (G >. But the G F as F is a upset. Now cosider F k = F ( [k+1] k. Agai, for all F Fk the set F belogs to F, thus F k is ( 1-almost itersectig ad the we are doe by Propositio

16 Suppose ow that = k is eve. The by Remark 5.1 ad Lemma 5. we ca assume that F is a upset ad all sets i F have size at least k 1. Furthermore Lemma 5. states that there is a upset F with F may members each of size at least k. This implies the boud i the theorem. Suppose F k 1, the just as i the case of odd, all sets of size at least k + 1 belog to F ad the proof of Lemma 5. shows that writig B k = {B ( [k] k \ Fk } we must have F k 1 = B k (as otherwise (F \ F k 1 B k would be a larger ( -almost itersectig family ad F k 1 = B k = {F ( [k] k 1 : B Bk (F B}. The Lovász versio [11] of the Kruskal-Katoa shadow theorem states that if B k = m = ( x k for some real umber x, the B k ( ( x k 1. As x ( k < x k 1 if x < k 1, we must have Fk 1 = B k ( k 1 k 1. Fially, ote that F k 1 is itersectig as a pair F,F F k 1,F F = would give, by the assumptio that F is a upset, D F (F {G : F G F } = 4. By the Erdős-Ko-Rado Theorem we obtai that F k 1 ( ( k 1 k < k 1 k 1. This cotradictio fiishes the proof of the theorem. Remark 5.6. For geeral l Lemma 5. gives that a ( l-almost itersectig family F of maximum size is a subset of {F [] : F ( l/ + 1}. Theorem 5.4 ca be used to give a better upper boud o the size of F. To see this ote that a ( l-almost itersectig family which is a upset satisfies the (1, p-chai itersectig property with p = log (l + 1. Ideed, if ot the there would exist a set F F ad a chai G 1 G... G p i F such that G p F =, but the all sets G with G 1 G G p would belog to F ad thus D F (F > l would hold. Refereces [1] R. Ahlswede, Simple hypergraphs with maximal umber of adjacet pairs of edges, J. Combiatorial Theory (B 8 (1980, [] A. Beráth, D. Gerber, Chai itersectig families. Graphs Combi. 3 (007, [3] B. Bollobás, O geeralized graphs, Acta Mathematica Academiae Scietarium Hugaricae 16 (1965 (34, [4] B. Bollobás, Sperer systems cosistig of pairs of complemetary subsets, Joural of Combiatorial Theory, Ser. A 15, [5] B. Bollobás, I. Leader, Set Systems with few Disjoit Pairs, Combiatorica 3 (003,

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