An intersection theorem for four sets

Transcription

1 An intesection theoem fo fou sets Dhuv Mubayi Novembe 22, 2006 Abstact Fix integes n, 4 and let F denote a family of -sets of an n-element set Suppose that fo evey fou distinct A, B, C, D F with A B C D 2, we have A B C D We pove that fo n sufficiently lage, F ( 1, with equality only if F F F This is closely elated to a poblem of Katona and a esult of Fankl and Füedi [10], who poved a simila statement fo thee sets It has been conjectued by the autho [18] that the same esult holds fo d sets (instead of just fou, whee d, and fo all n d/(d 1 This exact esult is obtained by fist poving a stability esult, namely that if F is close to ( 1 then F is close to satisfying F F F The stability theoem is analogous to, and motivated by the fundamental esult of Edős and Simonovits fo gaphs 1 Intoduction Thoughout this pape, X is an n-element set Fo any nonnegative intege, we wite ( X fo the family of all -element subsets of X In this pape we initiate a new appoach to solving classical intesection type poblems in extemal set theoy The appoach, which we call the stability method, poves an exact extemal esult by fist poving an appoximate esult that gives stuctual infomation on the nea extemal families Questions about stability in extemal combinatoics gew fom the seminal wok of Edős and Simonovits [23] on gaph stability in the 60 s The notion of stability fo popeties of set systems was explicitly fomulated by the autho ecently [17] Seveal motivations wee povided: Fist, Depatment of Mathematics, Statistics, & Compute Science, Univesity of Illinois at Chicago, 851 S Mogan Steet, Chicago, IL ; mubayi@mathuicedu Reseach patially suppoted by National Science Foundation Gant DMS , and an Alfed P Sloan Reseach Fellowship 2000 Mathematics Subject Classification: 05C35, 05C65, 05D05 Keywods: Extemal set theoy, intesecting family, stability 1

2 and pehaps the most obvious, is that poving a stability theoem tells us moe about a poblem than just the extemal esult Second, stability esults allow one to pove exact esults fo cetain extemal poblems This appoach was fist used by Simonovits [23] to detemine the exact extemal function fo k-citical gaphs, and moe ecently has found esugence in solving seveal hypegaph Tuán poblems (see [8, 9, 12, 15, 16, 21, 22] Thid, many poof techniques in extemal set theoy (Katona s cicle method, shifting, Linea Algeba methods seem not to give stability analogues, and theefoe new methods need to be developed Consequently, the seach fo stability esults can also yield new poof techniques in extemal set theoy Finally, stability esults can be used to accuately enumeate discete stuctues This was shown ecently by Balogh-Bollobás-Simonovits [1], who poved (among moe geneal statements that the numbe of labeled n vetex gaphs containing no copy of K l+1 is 2 (1 1/l(n 2+O(n 2 γ, whee γ > 0 This impoved pevious esults of Edős-Kleitman-Rothschild [4] and othes In the following definitions, we conside set-systems whose undelying set of elements ae not labeled, so technically speaking, a set-system efes to an isomophism class of set-systems whose undelying sets ae labeled A Popety P (P is an infinite family of set-systems (compising -sets The popety P is monotone if wheneve G P and G is obtained fom G by deleting vetices and edges, then G P One can chaacteize monotone popeties by popeties not containing fobidden subsystems In fact, the fobidden family fo P is the collection F of setsystems not contained in any membe of P The popety P n (Pn is the subfamily of P (P consisting of those set systems on n elements The classical extemal poblem in this egad is to detemine ex(n, F = max{ G : G P n }, o ex (n, F = max{ G : G Pn} whee F is the fobidden family fo a monotone popety P In what follows, we wite G n fo a set system whose undelying set has size n The fomulation below applies as well to P even though we wite it only fo P Definition Let t > 0 be an intege, P be a monotone popety of set systems, and F be a fobidden family fo P The popety P is t-stable if thee exists m 0 = m 0 (F and set systems Hm, 1, Hm t fo evey m > m 0 such that the following holds: fo evey δ > 0, thee exists ɛ > 0 and n 0 = n 0 (ɛ such that fo all n > n 0, if G n P n with G n > (1 ɛex(n, F, then G n can be tansfomed to some Hn i by adding and emoving at most δ G n sets Say that P is stable if it is 1-stable In [17] a stability theoem was poved fo a nontivial poblem in extemal set theoy, indeed this was one of the fist such esults Since we need this esult in ou poof, we descibe it 2

3 next A tiangle is a family of thee sets A, B, C that have paiwise nonempty intesections, and A B C = An old poblem of Edős was to detemine the maximum size of a family F ( X that contains no tiangle Extending pevious esults of Chvátal, Fankl, and Füedi, the autho and Vestaëte [19] poved that this maximum is ( fo all n 3/2 and 3 Recently, this was extended by the autho [17] to pove a stability vesion Theoem 1 ([17] Fix 3 Fo evey δ > 0, thee exist ɛ > 0 and n 0 = n 0 (ɛ, such that the following holds fo all n > n 0 : if G ( ( X contains no tiangle, G > (1 ɛ, then thee exists an S X with S = n 1 such that G ( ( S δ Hee we continue this poject, and pove a stability esult fo a genealization of a poblem of Katona and a theoem of Fankl and Füedi Moeove, ou appoach then yields an exact extemal esult Although the geneal technique of obtaining an exact esult afte obtaining stuctual infomation is not new (fo example, the delta system method initiated by M Deza is anothe example, to the authos knowledge, the stability appoach in this pape has not been peviously used to pove intesection theoems in extemal set theoy In paticula, sunflowes ae not employed in ou poof An ealie pape of Fankl and Füedi [11] also poves an exact esult fo an intesection theoem in seveal steps (building on the Deza technique, one of which obtains a stuctue theoem The eade may wish to compae and contast ou appoach to that in [11] A sta is a family of sets fo which thee is an element that is contained in all the sets The seminal esult of Edős-Ko-Rado [5] states that an intesecting family F ( X of maximum size is a sta fo n > 2 Motivated by a possible genealization of the Edős-Ko-Rado theoem to moe than two sets, Katona defined the following Definition Let s 3 Then f(n,, s denotes the maximum size of a family F ( [n] so that wheneve A, B, C F satisfy A B C s, we have A B C Katona asked fo the detemination of f(n,, s Fankl and Füedi [10] poved that fo evey 2 s 3, f(n,, s = ( as long as n 2 + 3, and obseved that f(n,, 2 1 = Ω(n fo fixed Note that the lowe bound f(n,, s ( is valid fo all s {2,, 3} by simply letting F be a maximum sized sta Moeove, by definition f(n,, s+1 f(n,, s, hence Fankl and Füedi s fist esult follows by poving the uppe bound just fo s = 2 They conjectued that f(n,, 2 = ( fo all 3 and n 3/2, with equality only fo a sta The theshold 3/2 follows fom the fact that fo smalle n, thee sets A, B, C F whose intesection is empty cannot exist (so in paticula, we can have F = ( n Fankl and Füedi [10] poved thei conjectue fo = 3, and commented (without poof that thei appoach also woks fo = 4, 5 and moe geneally fo > k 2 / log k Recently the autho [18] gave a shot poof of thei conjectue using diffeent aguments 3

4 Neithe of the two appoaches above povides a stability esult fo Katona s poblem, so we povide a thid appoach Although we couldn t show that ou method gives an exact esult fo all n 3/2, it poves a stability esult fo a moe geneal situation which includes Katona s poblem as a special case Moe pecisely, in Katona s poblem, the fobidden configuation is a family of thee sets with cetain popeties, while in ou esults it is a family of d sets with simila popeties Then we pove an exact esult (Theoem 3 using the stability theoem Fo convenience, we make the following Definition Fix 2 A family G of -sets is a K(d-family if, wheneve distinct sets A 1,, A d G satisfy i A i 2, we have i A i A K(2-family of -sets is simply an intesecting family, and hence its maximum size is given by the Edős-Ko-Rado theoem Also, f(n,, 2 is just the maximum size of a K(3-family of -sets on X Note that thee exist families of size n/ = Ω(n such that evey d 3 sets have empty intesection povided thei union is at most 2 1 (simply patition [n] into almost equal pats, and take all -sets with exactly one point in each pat This is the eason fo the theshold 2 in ou definitions As ou theoems below will show, changing the theshold 2 to any othe s 2 will not alte ou esults, simila to the situation egading f(n,, s descibed above (since the lowe bound ( on the families we conside holds fo all s 2 The stability esult below shows that if F is a K(d-family fo some 2 d, then F is stable Theoem 2 (Stability Fix 2 d Fo evey δ > 0, thee exists ɛ > 0 and n 0 such that the following holds fo all n > n 0 : Suppose that G ( ( X is a K(d-family If G (1 ɛ, then thee exists an (n 1-set S X with G ( ( S < δ ( 1 In paticula, G < (1 + δ It is possible that Theoem 2 holds even when d > In fact, it is an inteesting open poblem to detemine the lagest d = d( fo which Theoem 2 holds Using Theoem 2, we pove a esult simila to those of Fankl-Füedi [10] and the autho [18] fo K(4-families As mentioned befoe, ou poof technique fo Theoem 3 is one of the main new contibutions in this wok Theoem 3 Let 4 and let n be sufficiently lage Suppose that G ( [n] is a K(4-family Then G (, with equality only if F is a sta Theoem 3 is also elated to the following old poblem of Edős Let f (n be the maximum size of a family of -sets of an n element set containing no two pais of disjoint -sets with the same union Since all the fobidden configuations in this question ae fobidden configuations in a K(4-family (the convese is not tue, an uppe bound fo this poblem yields an uppe bound fo the K(4-poblem Answeing a question of Edős, Füedi [7] poved that f (n 7 ( n 2 4

5 The autho and Vestaëte [20] slightly impoved Füedi s esult by showing that f (n < 3 ( n Füedi futhe conjectued that f (n = ( 1 + fo all 4 and sufficiently lage n Theoem 3 can be viewed as a solution to a elaxation of this poblem (ignoing the (n 1/ tem The following moe geneal conjectue, posed in [18], emains open While a complete poof may be out of each at pesent, we cetainly believe that the stability appoach with Theoem 2 should yield a poof fo lage n Conjectue 4 ([18] Let d 3 and n d/(d 1 Suppose that G ( [n] is a K(d-family Then G (, with equality only if G is a sta It is possible that Conjectue 4 holds even fo d > Howeve, it cannot hold fo d 2, since in this case we can take an -patite -gaph G containing no copy of K(2,, 2, the complete - patite -gaph with two points in each pat It is known that such G exists with G > Ω(n 1+γ fo γ = 1 /2 1 > 0 (see [3, 6] and also [13] fo slight impovements, and it is easy to veify that G is a K(2 -family It would be inteesting to detemine the lagest d = d( so that evey K(d-family G ( X satisfies G = O(n 1 2 Notation Fo A ( X, let V (A = A A A Fo Y X, we define A Y = A ( X Y When Y = {y}, we wite A y instead of A {y} The tace of Y V (A in A is defined by t(y = t A (Y = {A X Y : A Y A} The degee of Y V (A in A is deg(y = deg A (Y = t A (Y When Y = {y}, we wite t(y and deg(y Let A ( X and x X Then we define S x = {Y t(x : deg(y = 1} and L x = t(x S x The sum of families A 1, A 2,, A t, denoted i A i, is the family of all sets in each A i Note that A i may have epeated sets, even if none of the A i have epeated sets The tace of A is t(a = x X t(x Wite S = x X S x and L = x X L x = t(a S Note that if A L x, then thee exists y x such that A L y The shadow G of a set system G ( X is G = {S ( X : thee exists T G with S T } Thoughout the pape, the Geek lettes ɛ, δ etc ae eal numbes and m, n,, s, t etc ae integes 3 Stability In this section we pove the stability esult fo those set systems which ae K(d-families fo some 2 d 5

6 Poof of Theoem 2 Fix 2 We poceed by induction on, handling the cases = 2 and = 3 sepaately When = 2, a K(2-family is a gaph containing no matching of size two, and in this case it is tivial to obseve that such a family with at least fou edges must be a sta So fo any δ > 0 (even δ = 0, we can let, fo example, ɛ = ɛ 2 = 1/8 and n 0 = 4 Indeed, then any gaph F on n > n 0 vetices with at least (1 ɛ ( = (7/8(n 1 4 edges must be a sta Moe geneally, a K(2-family is just an intesecting family It is well-known (see, eg, Theoem 2, page 48 of [2] that an intesecting family of size Ω(n is aleady a sta (indeed, this also follows fom the Hilton-Milne theoem on nontivial intesecting families, so a K(2-family is cetainly stable Consequently, we may assume that 3 d When = 3, a K(3-family contains no tiangle, since a tiangle A, B, C satisfies A B C =, and A B C A + ( B 1 + ( C 2 = 6 = 2 Hence Theoem 1 implies Theoem 2 fo = 3 We may theefoe assume that 4 Now suppose we ae given δ = δ as in the theoem Fist set { ( } 1 2 δ 1 = min 2 5 δ δ, (1 72( 1 Now choose ɛ 1 and n 0 (ɛ 1, 1 that satisfy the conclusion of the theoem fo 1 Such choices exist by the induction hypothesis, and we may also assume that ɛ 1 < δ 1 Next let ɛ = ɛ 1 2 (2 Finally, choose n 0 = n 0 (ɛ, > n 0 (ɛ 1, so that fo all n > n 0, (1 ɛ ( ( 1 n 1 > (1 2ɛ n ( n 2, (3 and ( 1 2 δ ( (n 2 2 ( n 2 > 5 δ (4 Note that a shot calculation shows that fo sufficiently lage n, both (3 and (4 do indeed hold, hence n 0 is well-defined Having fixed all constants, we now begin the agument fo the induction step As agued above, we may assume that d > 2, so fix 3 d Let G ( X be a K(d-family with X = n > n0 and G > (1 ɛ ( Ou stategy is to obtain the (n 1-set S in the conclusion of the theoem in thee steps: 1 Find a vetex w with L w vey lage 6

7 2 Study the stuctue of L w, in paticula, show that it contains a lage sta with cente x 3 Set S = X {x} and show that S satisfies the equiements of the theoem, because G is a K(d-family Step 1 We begin with the following equation which is an easy double counting execise G = deg(x = ( S x + L x = S x + L x x X x X x X x X Since x S x = S ( n, thee exists w X fo which L w G ( n 1 > (1 ɛ ( ( 1 n 1 n ( n ( n 2 n 2 > (1 2ɛ (1 ɛ 1, (5 whee the inequalities follow fom (3 and (2 This concludes Step 1 Step 2 Now conside the family L w ( X {w} 1 We next show that Lw is a K(d 1-family Suppose, fo a contadiction, that L w is not a K(d 1-family of ( 1-sets Then L w contains distinct sets A 1,, A d 1 with i A i 2( 1 and i A i = By definition of L w, thee exists y w such that A 1 {y} G Now define B i = A i {w} fo i = 1,, d 1 and B d = A 1 {y} Because i A i 2( 1, we have d i=1 B i = d 1 i=1 A i + {y, w} 2 If thee is an element v i B i, then v w, since w B d, and v y, since y B 1 Thus v i A i = which is impossible Consequently, i B i =, contadicting the fact that G is a K(d-family We conclude that L w is indeed a K(d 1-family By (5, we have L w > (1 ɛ 1 ( ( 1 ( 1 Because n0 > n 0 (ɛ 1, +1, and 2 d 1 1, the induction hypothesis applied to L w povides a vetex x X {w} so that L w ( X {w,x} 1 < ( δ n 2 1 Since ɛ 1 < δ 1, we conclude that ( ( n 2 n 2 deg Lw (x > (1 ɛ 1 δ 1 > (1 2δ 1 (6 This concludes Step 2 Step 3 The est of the poof is devoted to poving that G x = G ( X {x} satisfies ( n 1 G x δ (7 1 7

8 Patition G x into G 1 G 2, whee G 1 = {S G x : w S} and G 2 = {S G x : w S} We will sepaately bound the size of each of these families Let G w = t G1 (w In othe wods, { ( } X {w, x} G w = S : S {w} G 1 Note also that deg G1 (w = G 1 ( Claim 1 G 1 δ 2 ( Poof Suppose, fo a contadiction, that G 1 > δ 2 Then ( 1 deg Gw (T = G 1 > δ ( 1 2 T ( X {w,x} Consequently, thee exists T 0 ( X {w,x} fo which δ 2 ( 1 ( deg Gw (T 0 > ( n 2 1 = δ ( n (n 1 > δ (n 2 2 We will now obtain a contadiction to (6 Fist we show that thee is no E L w satisfying x E and E {x} t Gw (T 0 Suppose to the contay that such an E exists, say E = {x 1,, x, x} Since E L w, thee exists y {x, w} such that E {y} G Because δ (n 2 > 2, thee is an element z t Gw (T 0 E In paticula, T 0 {w, z} G Conside the d sets E {w}, E {y}, T 0 {w, z}, T 0 {w, x 1 },, T 0 {w, x d 3 } All these sets ae in G, and if d = 3, we conside only the fist thee Because T 0 E =, these thee sets have empty intesection On the othe hand, the union of these d sets is at most E + T = ( 1 + ( + 3 = 2 This contadicts the fact that G is a K(d-family Fom the above agument, we conclude that no E L w with x E satisfies E t Gw (T 0 Consequently, deg Lw (x ( n 2 ( deggw (T 0 < ( n 2 ( 1 2 δ ( (n 2 < 1 ( 2 5 δ (n 2, whee the last inequality follows fom (4 By the choice of δ 1 fom (1, this is uppe bounded by (1 2δ 1 ( n 2, a contadiction to (6 8

9 Befoe tuning ou attention to G 2, we need the following definition and esult Let { ( } X {w, x} G w,x = t G ({w, x} = E : E {w, x} G Claim 2 Thee ae disjoint ( 3-sets S 1, S 2 V (L w such that fo each i {1, 2}, {y V (L w {x} : S i {w, x, y} G} (1 4δ 1 (n + 1 Poof Let t be the numbe of ( 3-sets T V (L w {x} satisfying t Gw,x (T = {y V (L w {x} : T {w, x, y} G} (1 4δ 1 (n + 1 Since each set of t Lw (x contibutes to t Gw,x (T, when we sum we obtain ( (deg Lw (x = deg 3 Lw (x deg Gw,x (T This implies that By (6, this is at least T ( V (Lw {x} 3 t(n t (deg L w (x ( n 2 3 (1 4δ 1 (n + 1 4δ 1 (n + 1 [( ] n 2 t (1 4δ 1 (n ((1 2δ 1 ( ( n 2 n 2 3 (1 4δ 1 (n + 1 4δ 1 (n + 1 = 1 2δ ( 1 n 2 4δ 1 3 = 1 ( n ( n 3 > = 4 1 4δ 1 4δ 1 ( (n 2 1, ( 3 1 ( n 2 3 whee the last inequality holds since n > n 0 > 2 Thus the Edős-Ko-Rado theoem applies to give the sets S 1 and S 2 Define, fo i {1, 2}, A i = {y V (L w {x} : S i {w, x, y} G}, and let A = A 1 A 2 = {y V (L w : S i {w, x, y} G fo i = 1, 2} 9

10 Set B = X A {w, x} Note that S 1 S 2 B By Claim 2, A i (1 4δ 1 (n + 1 fo i = 1, 2 Theefoe A = A 1 A 2 2(1 4δ 1 (n + 1 n It now follows that B n A 2n 2(1 4δ 1 (n + 1 9δ 1 n By adding points fom A abitaily to B, we may assume that B = 9δ 1 n Claim 3 G 2 18δ 1 n ( Poof Patition G 2 into G 21 G 22, whee G 21 = {E G 2 : E B 1} and G 22 = {E G 2 : E B 2} Subclaim 31 G 22 < 2 B ( Poof In what follows, the families S y and L y ae taken with espect to G 22 Since each set in G 22 contains at least two elements in B, 2 G 22 y B deg G22 (y = y B ( S y + L y = y B S y + y B L y (8 Recall that S B = y B S y G 22 The definition of G 22 implies that evey E S B satisfies E B 1 Moeove, fo evey E S B, thee is exactly one y fo which E S y Theefoe { ( } X S B E : E B 1, 1 and consequently ( n 1 S y = S B B (9 y B Since G 22 is a K(d-family, the agument in Step 2 implies that L y is a K(d 1-family fo evey y B Since n 0 1 > n 0 (ɛ 1, 1, and 2 d 1 1, the induction hypothesis applies to L y and L y (1 + δ 1 ( ( n 2 < 2 n 2 fo evey y B Theefoe ( n 2 L y < 2 B (10 y B Now (8, (9 and (10 imply ( n 1 2 G 22 < 3 B 10

11 This gives the equied bound (with oom to spae on G 22 Togethe with the fact that B = 9δ 1 n, Subclaim 31 implies that ( n 1 G 22 < 18δ 1 n (11 Subclaim 32 G 21 = Poof Suppose on the contay that E G 21 Since E B 1 and S 1 S 2 B, we have E (S 1 S 2 1 Futhemoe, by Claim 2, we know that S 1 and S 2 ae disjoint Theefoe E has empty intesection with at least one S i, say S 1 Since d, we may choose d 1 distinct elements y 1,, y d 1 A E By the definition of A, we know that A i = S 1 {w, x, y i } G fo all i = 1,, d 1 The d 1 sets above togethe with A d = E yield d sets A 1,, A d G Because E (S 1 {w, x} =, it is easy to see that d i=1 A i = Also, d i=1 A i = E + S 1 +{w, x} = 2 1 This contadicts the fact that G is a K(d-family Subclaim 32 and (11 yield G 2 < 18δ 1 n (, which finishes the poof of Claim 3 Since n > 2 and δ 1 δ /[72( 1] by (1, ( n 1 18δ 1 n = 18δ 1 n 1 ( ( n 1 n 1 < 36δ 1 ( 1 n Consequently, Claims 1 and 3 give G x = G 1 + G 2 δ 2 and the poof of the theoem is complete ( n 1 + δ 1 2 < δ 2 ( ( n 1 n 1 = δ, 1 1 ( n Fom stability to an exact esult In this section we use the stability esult poved in the last section to give the exact esult in Theoem 3 fo lage n Poof of Theoem 3 Let b 0 = b 0 ( be the theshold fom Theoem 2 with d = 4 and δ = 1 In othe wods, evey K(4-family on b b 0 vetices has size at most 2 ( b 1 We may also assume that b 0 > 10 2 (12 11

12 Now select ( 1! δ = 4 (13 Let n 0 be the output fom Theoem 2 with d = 4 fo this δ (Theoem 2 also outputs ɛ but this is not elevant fo us Finally, choose N so that N > max{2 3, n 0 } and ( N 1 > 2b 0 (14 Suppose that n > N and G ( ( X is a K(4-family ( X = n with G = We will show that G is a sta Since a sta is a maximal K(4-family, this poves the equied bound on G, with the chaacteization of equality as well As n > N > n 0, thee exists x X such that ( n 1 m := G x < δ (15 1 If m = 0, then G is a sta and we ae done, hence we may assume that m > 0 Let G x = t G (x = { ( } X E : E {x} G 1 Since each set in G x coesponds bijectively to an edge containing x, ( n 1 G x = m (16 1 Claim 1 Thee ae paiwise disjoint (-sets S 1, S 2,, S ( X {x} such that deg Gx (S i = {y X : S i {x, y} G} n + 1 2m ( Poof Let t be the numbe of (-sets T X {x} satisfying Then ( 1 G x = ( 1 G x = deg Gx (T n + 1 2m T ( X {x} deg Gx (T ( (( ( n 1 t(n t n + 1 ( 2m 12

13 This implies that t 2m ( 1 G x ( ( ( n 1 n + 1 2m ( By (16, the RHS is ( 1 (( n 1 m 1 ( ( n 1 n + 1 2m ( = ( 1m + 2m Hence t ( ( n 1 > 1 2 ( n 1 Now conside the family of all (-sets descibed above, and let T 1,, T l be a maximum matching in this family If l <, then all othe sets of this family have an element within i T i, which implies that t ( 1( ( 3 < 2 ( ( 3 < /2, because n > 2 3 fom (14 This contadiction shows that l and the claim is poved By Claim 1, fo evey 1 i {y X : S i {x, y} G} < + 2m ( Let B = {y X : S i {x, y} G fo some i []} Then B < m/ ( By adding points abitaily to B, we may assume that Now define, fo each i {0,, }, B = m + (17 ( T i = {T G x : T B = i} Note that T 0 T is a patition of G x In the emainde of the poof, we will show that G x = i=0 T i < m, theeby contadicting (15 We fist need two moe Claims Claim 2 T p = fo 0 p < 1 Poof If T T p, then wite T = T 1 T 2, whee T 1 = T B (so T 1 = p and T 2 = T T 1 Since T 1 < 1, and the sets S 1,, S ae paiwise disjoint, we may assume that S i T 1 = fo some i (note also that each S j B Since p, we have T 2 2 Let y, z, w be thee elements outside B, at least two of which ae in T 2 Then S i {x, y}, S i {x, z}, S i {x, w} ae 13

14 all sets in G Togethe with T this yields fou sets whose union is at most 2 and intesection is empty This contadicts the hypothesis that G is a K(4-family Claim 2 implies that G x = T T 1 We next estimate the size of T 1 Claim 3 T 1 ( B Poof Suppose thee exists an ( 1-set E B and elements y, z B such that E {y}, E {z} T 1 Since E = 1, as befoe we may assume that S i E = fo some i By Claim 1, we have S i {x, y}, S i {x, z} G Togethe with E {y} and E {z}, this yields fou sets in G whose union is 2 and intesection is empty This contadicts the fact that G is a K(4-family Consequently, we may count sets in T 1 by thei intesection with B This yields T 1 ( B We now conside two cases, depending on the size of B Case 1: B < b 0 Clealy T and can be bounded by ( B Togethe with Claim 3, this gives ( ( B B m = G x = T + T 1 + < 2 B < 2b 1 0 Call each -set in G x bad, and each -set containing x that is absent fom G missing Thus both the numbe of bad edges and missing edges is m Now pick a bad edge S Then fo each (-set E X {S {x}}, at least one of the -sets E {x, v} fo v S must be missing, since othewise thee of these sets togethe with S would imply that G is not a K(4-family Consequently, to each bad set we may associate ( n 1 missing sets By (14, this is aleady geate than 2b 0 > m, and so we can have no bad sets at all Thus in this case m = 0 which is a contadiction Case 2: B b 0 In this case, the agument woks because we can bound the size of T Since T is itself a K(4-family, the choice of b 0 implies that T 2 ( B 1 1 Recalling fom (17 that B = 2 2 m/ ( + 2, and using B b 0 > 10 2 fom (12, we obtain B < 3 2 m/ ( Consequently, ( ( B 1 B m = G x = T + T 1 < 2 + < 3 B ( 1! < 2 m 1 ( 1 Simplifying, m > ( n > ( n 1 (( 1 (n 1(( > 2 ((+2 This implies that (n 1 1 (n 1 1 m > > 1+2/( 4 On the othe hand, by (15 we know that m < δ ( < δ(n 1 1 /( 1! Putting these togethe yields δ > ( 1!/ 4 which contadicts (13 and completes the poof 14

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