O adomly geeated o-tivially itesectig hypegaphs Balázs Patkós Submitted: May 5, 009; Accepted: Feb, 010; Published: Feb 8, 010 Mathematics Subject Classificatio: 05C65, 05D05, 05D40 Abstact We popose two pocedues to choose membes of sequetially at adom to fom a o-tivially itesectig hypegaph. I both cases we show what is the limitig pobability that if = c 1/3 with c c, the the pocess esults i a Hilto-Mile-type hypegaph. 1 Itoductio I 1961, Edős, Ko ad Rado [5] poved that if, the the edge set E of a itesectig -uifom hypegaph with vetex set V ad V = caot have lage size tha 1 1, moeove if <, the the oly hypegaphs with that may edges ae of the fom {e V : v e} fo some fixed v V. I the past almost five decades, the aea of itesectio theoems has bee widely studied, but adomized vesios of the Edős-Ko-Rado theoem have oly attacted the attetio of eseaches ecetly. Thee ae maily two appoaches to the adomized poblem. Balogh, Bohma ad Mubayi [] cosideed the poblem of fidig the lagest itesectig hypegaph i the pobability space G, p of all labeled -uifom hypegaphs o vetices whee evey hypeedge appeas adomly ad idepedetly with pobability p = p. I this pape, we follow the appoach of Bohma et al. [3], [4]. They cosideed the followig pocess to geeate a itesectig hypegaph by selectig edges sequetially ad adomly. Choose Radom Itesectig System Choose e 1 uifomly at adom. Give Fi = {e 1,..., e i } let AF i = {e : e / F i, 1 j i : e e j }. Choose e i+1 uifomly at adom fom AF i. The pocedue halts whe AF i = ad F = F i is the output by the pocedue. Depatmet of Compute Sciece, The Uivesity of Memphis, Memphis, TN, 3815, USA. Suppoted by NSF Gat #: CCF-07898. E-mail: bpatkos@memphis.edu, patkos@eyi.hu the electoic joual of combiatoics 17 010, #R6 1
Theoem 1.1 Bohma et al. [3] Let E, deote the evet that F = 1 1. The if = c 1/3, lim PE, = 1 if c 0 1 1+c 3 if c c 0 if c. Theoem 1.1 states that the pobability that the esultig hypegaph will be tivially itesectig i.e. all of its edges will shae a commo elemet with pobability tedig to 1 i othe wods, with high pobablity, w.h.p. povided = o 1/3. I this pape we will be iteested i two pocesses that geeate o-tivially itesectig hypegaphs fo this age of. Befoe itoducig the actual pocesses, let us state the theoem of Hilto ad Mile that detemies the size of the lagest o-tivially itesectig hypegaph. Theoem 1. Hilto, Mile [6] Let F be a o-tivially itesectig hypegaph with 3, + 1. The F 1 1 1 1 + 1. The hypegaphs achievig that size ae i fo ay -subset F ad x \ F the hypegaph F HM = {F } {G : x G, F G }, ii if = 3, the fo ay 3-subset S the hypegaph F = {F 3 : F S }. We will call the hypegaphs descibed i i HM-type hypegaphs, while hypegaphs F fo which thee exists a 3-subset S of such that F cosists of all -subsets of with F S will be called -3 hypegaphs eve if > 3 the atual geealizatios of hypegaphs of the fom of ii. We ow itoduce the two pocesses we will be iteested i. I some sese they ae the opposite of each othe as the fist pocess assues as ealy as possible i.e. whe pickig the thid edge e 3 that it poduces a o-tivially itesectig hypegaph while the secod oe is the same as the oigial pocess of Bohma et al. as log as it is possible that the pocess esults a o-tivially itesectig hypegaph. The mai value of the fist model is that the esults coceig this model allows us to calculate the pobability that the oigial model of Bohma et al. poduces a HM-type hypegaph whe = Θ 1/3, while the secod model seems to be the model that ca be obtaied with the least modificatio to the oigial such that it esults a o-tivially itesectig hypegaph fo all values of ad. Hee ae the fomal defiitios. The Thid Roud Pocess Choose e 1 uifomly at adom. Give Fi = {e 1,..., e i } if i let AF i = {e : e / Fi, 1 j i : e e j } while fo i = let AF = {e : e / F, e e j j = 1,, e e 1 e = }. Choose e i+1 uifomly at adom fom AF i. The pocedue halts whe AF i = ad F = F i is the output by the pocedue. Note that by Lemma 7 i [3] if O /3 = = ω 1/3, the w.h.p. F 3 of the oigial pocess of Bohma et al. is o-tivially itesectig ad thus the two pocesses ae the the electoic joual of combiatoics 17 010, #R6
same w.h.p. The pobability of a evet E i the Thid Roud Pocess will be deoted by P 3R E. The Put-Off Pocess Choose e 1 uifomly at adom. Give Fi = {e 1,..., e i } let { } AF i = e : e / F i ; G o-tivially itesectig with {e} F i G, i.e. AF i is the set of all edges that ca be added to F i such that {e} F i ca be exteded to a o-tivially itesectig hypegaph. Choose e i+1 uifomly at adom fom AF i. The pocedue halts whe AF i = ad F = F i is the output by the pocedue. Note agai that by Lemmas 7 ad 8 i [3] if = ω 1/3, the w.h.p aleady F log of the oigial pocess of Bohma et al. is o-tivially itesectig ad thus the two pocesses ae the same. The pobability of a evet E i the Put-Off Pocess will be deoted by P PO E. If the pobability of a evet E is the same i the two models o the same boud applies fo it i both models, the we will deote this pobability by P 3R,PO E. The pobability of a evet E i the oigial pocess will be deoted by P INT E. To fomulate the mai esults of the pape we eed to itoduce the followig evets: E HM stads fo the evet that the pocess outputs a HM-type hypegaph while E deotes the evet that the output is a -3 hypegaph. Theoem 1.3 If ω1 = = c 1/3, the lim P 3RE HM = Theoem 1.4 If 3 is a fixed costat, the 1 lim P 3RE HM = 1 1 Theoem 1.5 If = c 1/3, the lim P POE HM = 1 if c 0 1 1+c 3 /3 if c c 0 if c. 3, lim P 3RE = 1 1+c 3 + c3 Coollay 1.6 If = c 1/3 with c c, the P INT E HM = 1 if c 0 1 if c 1+c 3 /3 c 0 if c. 1+c 3 c3 1 + c 1 3 1 + c 3 /3. 3 1. 1 The est of the pape is ogaized as follows: i the ext sectio we itoduce some evets that will be useful i the poofs ad estate some of the lemmas of [3]. I Sectio 3, we pove Theoem 1.3 ad Coollay 1.6, Sectio 4 cotais the poof of Theoem 1.4 ad Sectio 5 cotais the poof of Theoem 1.5. the electoic joual of combiatoics 17 010, #R6 3
Defiitios ad Lemmas fom [3] g We will wite g = of g = ωf to deote the fact that lim = 0 f =, while g = Of g = Ωf will mea that thee exists a lim g f positive umbe K such that g < K g > K fo all iteges ad g = Θf f f deotes the fact that both g = Of ad g = Ωf hold. Thoughout the pape log stads fo the logaithm i the atual base e. We will use the followig well-kow iequalities: fo ay x we have 1 + x e x ad if x teds to 0, the 1 + x = expx + Ox. Biomial coefficiets will be bouded by a b b a b ea b b. Fially, fo biomial adom vaiables we have the followig fact see e.g. [1]. Fact.1 If X is a adom vaiable with X Bi, p, the we have P X p > δp e δ p/3. I paticula, fo ay costat c with 0 < c < 1 we have P X p > cp = exp Ωp. We call a hypegaph with i edges a i-sta if the paiwise itesectios of the edges ae the same ad have oe elemet which we will call the keel of the i-sta. A hypegaph of 3 edges e 1, e, e 3 is a tiagle if 3 i=1 e i = ad e i e j = 1 fo all 1 i < j 3. The base of a tiagle is the 3-set {e i e j : 1 i < j 3}. A hypegaph is a suflowe if the itesectio of ay two of its edges ae the same which is the keel of the suflowe. A hypegaph H of 3 edges is a -tiagle if H ca be patitioed ito 3 suflowes each of edges with keel size such that ay 3 edges take fom diffeet suflowes fom a tiagle with the same base. A hypegaph of edges e 1 1, e 1,..., e1, e is a -double-boom if i=1 j=1 ej i = 1, e 1 i e i = fo all 1 i ad ej i ej i = 1 fo ay i i. We call i=1 j=1 ej i the keel of the double-boom. The subhypegaph of a -double-boom cosistig of the d + edges e 1 1, e 1,..., e1 d, e d, e1 d+1,..., e1 is a d-patial -double-boom. The elemets ot idetical to the keel that belog to e 1 i e i ae called the semi-keels of the d-patial -double-boom ad the sets e 1 j without the keel d + 1 j ae called the loely figes of the d-patial -double-boom. The followig two tivial popositios show what itesectig subhypegaphs of F j assue that the output of the pocess will be -3-hypegaph o a HM-type hypegaph. Popositio. If a itesectig hypegaph H cotais a -tiagle, the thee is oly oe maximal itesectig hypegaph H cotaiig H ad H is a -3-hypegaph. Popositio.3 If a -set f does ot cotai the keel x of a d-patial -doubleboom B, but meets all sets i B, the f must cotai all semi-keels of B ad meet each loely fige of B i exactly oe elemet. I paticula, the oly -set meetig all sets of a -double-boom ot cotaiig the keel is the set of all semi-keels ad thus the electoic joual of combiatoics 17 010, #R6 4
Figue 1: A -tiagle with = 3. if a itesectig hypegaph H cotais a -double boom, the thee is oly oe otivially itesectig hypegaph H that cotais H ad H is a HM-type hypegaph. Let A i be the evet that F i is a i-sta. Let A j, deote the evet that F j cotais a -sta ad thee exists at most 1 edge e F j ot cotaiig the keel of the -sta. I paticula, A, = A. Let A j, deote the evet that F j cotais a -double boom ad thee exists at most 1 edge e F j ot cotaiig the keel of the -double boom. Let H deote the evet that e 3 cotais all of e 1 e as well as at least oe vetex fom e 1 \ e e \ e 1. Let deote the evet that F 3 is a tiagle. Let j,l deote the evet that F j cotais a l-tiagle ad all edges i F j meet the base of this l-tiagle i at least elemets. Let B j deote the evet that e F j e. Let C j,1 deote the evet that F j+1 is a j-sta with a tasvesal, a set meetig all sets of the sta i 1 elemet which is diffeet fom the keel of the sta. the electoic joual of combiatoics 17 010, #R6 5
Figue : A -double boom with = 4. Let C l,j,1 deote the evet that F l cotais a j-sta T ad thee is exactly oe edge e F l ot cotaiig the keel of T ad e is a tasvesal of T. I paticula, C j+1,j,1 = C j,1. Let C l,j,1 deote the evet that F l cotais a subhypegaph H of a -double boom B with EH = j ad thee is oly oe edge e F l ot cotaiig the keel of H ad e is the set of semi-keels of B. Let D j deote the evet that thee exists a x such that thee is at most oe edge e F j that does ot cotai x. Fo ay evet E, the complemet of the evet is deoted by E. We fiish this sectio by statig some of the lemmas fom [3] that we will use i the poofs of Theoem 1.3 ad Theoem 1.5. Lemma.4 Lemma 1 i [3] If = o 1/, the w.h.p. A holds. Lemma.5 Lemma i [3] If = o 1/, the P INT A 3 = 1 o1 1 + 13 1 + o1. the electoic joual of combiatoics 17 010, #R6 6
Lemma.6 Lemma 3 i [3] If = o /5 ad m = O 1/ /, the m PA m A 3 = exp + o1. 4 Lemma.7 Lemma 4 i [3] If = o 1/, the PH A = o1. Lemma.8 Lemma 7 i [3] If ω 1/3 = = o /3, the PB 3 = o1. 3 The Thid Roud Model I. I this sectio we pove Theoem 1.3 ad Coollay 1.6. Fist we give a outlie of the poof, the we poceed with lemmas coespodig to the diffeet cases of Theoem 1.3 ad at the ed of the sectio we show how to deduce Theoem 1.3 fom these lemmas ad how Coollay 1.6 follows fom Theoem 1.3 ad Theoem 1.1. Outlie of the poof : We will use Popositio. ad Popositio.3 to calculate the pobability of the evets E ad E HM, while to pove that E HM does ot hold w.h.p. if = ω 1/3 we will show that fo evey vetex x thee exist at least edges i F i oe of them cotaiig x, i.e. D i does ot hold. The latte will be doe by Lemma 3.4 ad Lemma 3.5. To show the emegece of a -double boom we will pove i Lemma 3.3 that it follows fom the ealy appeaace of a 3-sta of which the pobability is calculated i Lemma 3.. Ou fist lemma states that if = o 1/, the F 3 is a tiagle w.h.p. Lemma 3.1 I the Thid Roud Model, if = o 1/, the holds w.h.p. Poof. 1 3 3 3 P 3R A 1 +1 = O = O exp 4 3 3 + 5 Togethe with Lemma.4, this poves the statemet. 3 j + 1 j = o1. 4 j=0 Lemma 3., fo the Thid Roud Model, is the equivalet of Lemma.5 i [3] fo the Itesectio Model. It gives the pobability that F 4 cotais a 3-sta. Lemma 3. I the Thid Roud Model, if = o 1/, the P 3R C 3,1 = 1 o1 1 + 1 + 3 3. the electoic joual of combiatoics 17 010, #R6 7
Poof. If S is the base of F 3, the the keel of the 3-sta i F 4 ca oly be a elemet of S. Thus the umbe of sets that ca exted F 3 to F 4 i such a way that C 3,1 should hold is 3 3+3. Let Ni deote the umbe of sets f i AF 3 with f S = i i = 0, 1,, 3. Evey set f with f S belogs to AF 3, sets belogig to AF 3 with f S = 1 must meet oe edge of F 3 outside S, while sets disjoit fom S that belog to AF 3 must meet all thee edges i F 3 outside S. Theefoe we have the followig bouds o N i : 3 3 N = 3 3, N 3 =, 3 1 4 3 N 1 3, 3 + 3 6 3 N 0 3. 3 3 By the assumptio = o 1/ we have c 1 c expo 1 fo ay costats c 1, c, ad thus the lowe ad uppe bouds o N 0 ad N 1 ae of the same ode of magitude. Hece we obtai P 3R C 3,1 = 3 3+3 3 i=0 N i 3 3+3 = 3 3 3 + 3 3 + 3 4 + 3 6 3 1 + o1 1 =. 1 3 + o1 + 1 + 1 + o1 3 Lemma 3.3 states that if F j cotais a 3-sta fo some small eough j, the F will cotai a -double boom w.h.p. which by Popositio.3 assues that the pocess outputs a HM-type hypegaph. Lemma 3.3 If = O 1/3 ad j log, the P 3R l : A l, C j,3,1 = 1 o1. Poof. Suppose C j,3,1 holds fo some j with j j log. The the umbe of sets i AF j cotaiig the keel of a 3-sta S i F j is 1 1 M = j + 1 1 1 as they all must meet the tasvesal t of S aleady i F j. Clealy, we have 1 j + 1 M, the electoic joual of combiatoics 17 010, #R6 8
as fo the lowe boud we eumeated the -sets cotaiig the keel ad exactly oe elemet of t, while fo he uppe boud we couted times the umbe of -sets cotaiig the keel ad oe fixed elemet of t. The umbe of sets i AF j ot cotaiig the keel of S is at most 5 4 3 3 + 3 1, 1 4 3 whee the fist tem of the sum stads fo the sets i AF j that meet all elemets of S outside t ad thus we have to make sue that they meet t as well, while the secod tem stads fo the othe sets. Thus the pobability that the adom pocess picks a edge ot cotaiig the keel is at most 3 3 5 4 + 3 1 4 3 1 45 j + 1 4 5 = O 1. + 6 + 6 1 Remembe that D k deotes the evet that thee is a vetex x which is cotaied i all but at most oe edge of F k, thus as = O 1/3, we obtai that P PO,3R D 1/7 C j,3,1 = 1 o1. Fo i j let α i deote the maximum umbe k such that thee exist k edges i F i that fom a subhypegaph of a -double boom of which the semi-keels ae elemets of t, i paticula α i = implies the existece of a -double boom. Let us itoduce the followig adom vaiables: { 1 if αi α Z i = i+1 o α i = 0 othewise. The umbe of edges that would make α i gow if α i < is at least α i αi 1. The total umbe of edges i AF i is at most 1 + 1 3 4 3 = O 1 as = O 1/3. Thus fo j i 1/7 we have αi α i 1 P 3R,PO Z i = 1 D 1/7, C j,3,1 = Ω αi = Ω 1 1 3 αi = Ω as = O 1/3. Note that if α i =, the by defiitio PZ i = 1 = 1, thus ay lowe boud obtaied i the α i < case is valid i this case, too. Let us coside cases: Case I = o 1/15 By, we have P 3R Z i = 1 D 1/7, C j,3 Ω1/, thus 1/7 P 3R Z i < D 1/7, C j,3,1 < PBi 1/7, Ω1/ < 0 i=j the electoic joual of combiatoics 17 010, #R6 9
as 1/7 = ω by the assumptio = o 1/15. Case II = ω 1/16 By, we obtai P 3R Z i = 1 D 1/7, C j,3,1, α i / = Θ1, thus P 3R 1/7 Z i < 1/0 D 1/7, C j,3,1 < PBi 1/7, Θ1 < 1/0 0 i=j as 1/0 < / by the assumptio = ω 1/16. Fo ay subhypegaph of a -double boom thee exists a set of at least half the edges that ae paiwise disjoit apat fom the keel, thus if α i 1/0, the the umbe of - sets that do ot cotai the keel but meet all edges of F i is at most 1 1/0 1/0. 1/0 As befoe, if thee is oly oe edge i F i ot cotaiig x, the the umbe of -sets i AF i cotaiig x is 1 1 1 1 j + 1 1. Hece, we have 1/0 11/0 1/0 P 3R D C 1/7, 1/0,1 1 1/0 0 as = o 1/ ǫ. O the othe had, just as i we have P 3R Z i = 1 D, C j,3,1 = Ω αi = Ω1/ ad thus P 3R Z i < D, C j,3,1 PBi, Ω1/ < 0 i=j as / = ω sice = O 1/3. Lemma 3.4 assets that if ω 1/3 = = o 1/ log 1/10, the all vetices ae cotaied i at most edges of F 4 ad theefoe the esultig hypegaph of the pocess caot be HM-type. Lemma 3.4 If ω 1/3 = = o 1/ log 1/10, the P 3R,PO D 4 = o1. the electoic joual of combiatoics 17 010, #R6 10
Poof. We coside cases: Case I ω 1/3 = = o 1/ I this case, Lemma 3.1 states that holds w.h.p., ad the computatio i Lemma 3. shows that e 4 is disjoit fom the base of the tiagle of F 3 w.h.p. Case II ω 1/ log 1 = = o 1/ log 1/10 Fist ote that 1/ 1/ 3 3 1/ exp O Usig 3 ad witig E fo the evet e 1 e 1/ we have P 3R E 1/ 1/ 1/ 1 = e 1/ 1/ 1/. 3 exp O, as the deomiato bouds fom below the umbe of -sets meetig e 1 i exactly 1 elemet, while the eumeato is a uppe boud o the umbe of -sets meetig e 1 i at least 1/ elemets. This boud teds to 0 as 3/ / teds to 0. Futhemoe, still usig 3 ad witig E 3 fo the evet e 1 e e 3 1/, we have 1/ P 3R E 3 E 1/ 1/ 1/ = e 1/ 1/ 1/ exp O, which teds to 0 fo the same easo as the pevious boud. Now ote that by the defiitio of the Thid Roud pocess e 1 e e 3 = ad thus D 4 is equivalet to e 4 1 i<j 3 e i e j. As E, E 3 imply e 1 \ e e 3, e \ e 1 e 3, e 3 \ e 1 e 1/, we have 1/ P 3R D 4 E, E 3 1/ 3 3 = 3 5/ exp O = O 6/5 5/4 log 3 0, whee fo the last equality we used the assumptio ω 1/ log 1 = = o 1/ log 1/10 to obtai exp O = O 1/5 ad 5/ = Ω 5/4 log 5/. Lemma 3.5 is the equivalet of Lemma 8 i [4] ad the poofs ae almost idetical. Lemma 3.5 If = ω 1/ ad log m exp, the 3 P 3R,PO D m = o1. Poof. Pick the fist 3 edges e 1, e, e 3 accodig to ay of the pocesses ad the coside m elemets of \ {e1, e, e 3 } beig chose at adom without eplacemet. The pobability that these m + 3 edges fail to fom a itesectig family is at most m 3m + m exp 1 exp, 3 the electoic joual of combiatoics 17 010, #R6 11
which teds to 0 as = ω 1/. We kow that the Put-Off pocess ad the Radom Itesectig Hypegaph pocess is w.h.p. the same if = ω 1/3 ad the Thid Roud pocess is the same as the Radom Itesectig Hypegaph pocess fom the fouth oud by defiitio. Thus coditioig o e 1, e, e 3, the distibutio of F m+3 will be the same as pickig m distict -sets uifomly at adom if we futhe coditio o the evet of pobability 1 o1 that the adomly picked sets togethe with the fist 3 edges fom a itesectig hypegaph. Thus we obtai that P 3R,PO D m 1 exp + o1 + m 3 1 1 m 1 = O exp + o1 + m m 1 m O exp + o1 + m m. 3 3 Poof of Theoem 1.3. We coside seveal cases. Let ω1 = = c 1/3. If c teds to 0, the fom Lemma 3.1 ad Lemma 3. it follows that C 3,1 holds w.h.p ad the Lemma 3.3 togethe with Popositio.3 fiishes the poof of this case. If c c, the Lemma 3.1 states that holds almost suely. Accodig to Lemma 3. the pobability that C 3,1 holds is 1 1+c 3 1 + o1 ad the poof of Lemma 3. shows that if C 3,1 does ot hold, the o does D 4 thus E HM caot happe. Agai, Lemma 3.3 togethe with Popositio.3 fiishes the poof of this case. Fially, if c teds to ifiity the fo ω 1/3 = = o 1/ log 1/10 Lemma 3.4 while fo ω 1/ = / Lemma 3.5 poves that the pobability of E HM is o1. Poof of Coollay 1.6: Bohma et al. i [3] pove that coditioed o the evet A 3, a tivially itesectig family is the output of the oigial pocess w.h.p. Lemma.4 ad Lemma.7 give that coditioed o the evet that A3 does ot hold, we have e F 3 e = with pobability tedig to 1, that is, e 3 is chose accodig to the ule of the Thid Roud Pocess. Thus the followig equality holds: lim P INTE HM = 1 lim P INT A 3 lim P 3R E HM. Lemma.5 ad Theoem 1.3 complete the poof. 4 The Thid Roud Model II. costat I this sectio we coside the Thid Roud Model whe is a fixed costat ad we pove Theoem 1.4. We will use oe of the lemmas poved i the pevious sectio ad we will also eed ew oes. Lemma 4.1 states that F log cotais eithe a 3-sta o a -tiagle w.h.p. but ot both. the electoic joual of combiatoics 17 010, #R6 1
Lemma 4.1 If 3 is a costat, the 3 1 P 3R log, = 1 + o1, 1 3 1 P 3R C log,3,1 = 1 1 + o1, 1 P 3R log, C log,3,1 = o1. Poof. Let S be the base of F 3 if holds ad fo ay 3 j log ad i = 0, 1,, 3 let AF j,i deote the -sets i AF j that meet S i i elemets ad let M j,i = AF j,i. We will pove that fo ay 3 j log the pocess picks a edge eithe fom AF j,1 o fom AF j, w.h.p. Futhemoe, if fo some 3 j log the pocess picks a edge fom AF j,1, the C log,3,1 holds w.h.p., while if this is ot the case i.e. the pocess picks a edge fom AF j, fo all 3 j log, the log, holds w.h.p. We will eed some bouds o M j,i. Clealy, we have 3 6 M j,3, M j,0 3. 3 3 Fo the fist iequality we couted all -sets cotaiig S, while fo the secod iequality we used that if a -set f AF j is disjoit fom S, the f must meet e 1, e, e 3 outside S. Thus as is costat we have M j,0, M j,3 = O 3. Let j deote the evet that fo all edges e of F j we have e S. Clealy 3 holds. If j holds, the evey edge f with f S = belogs to AF j, theefoe we have M j, 3 3 j = Θ. By compaig this to the bouds o M j,0 ad M j,3 we obtai that if thee is a j log such that e j+1 / AF j,, the e j+1 AF j,1 w.h.p. We claim that if j holds, the all edges of F j ae paiwise disjoit outside S w.h.p. Ideed, fo ay 3 j j the umbe of -sets f AF j that meet e Fj e \ S ad f S = is at most 3 4 3 log = O 3 log as we ca choose i 3 ways which elemets of S belog to f, which othe elemet of e Fj e \ S belogs to f ad e Fj e\s log as j log. Thus the pobability that e j +1 meets e Fj e\s fo some j log is Olog /. Thus if j deotes the evet that j holds ad the edges i F j ae paiwise disjoit outside e Fj e\s, the we have just see that j implies j w.h.p. To calculate the pobability that thee is a j log fo which e j+1 AF j,1 ad to have moe isight o the pocess we itoduce some moe otatios. Let S j = {s S : d j s = j 1} ad h j = S j, i.e. S j is the set of elemets which ae cotaied i all but at most oe edge of F j ad theefoe they still might become the keel of a possible HM-type extesio of F j. As holds, we have S 3 = S ad h 3 = 3. Note that if fo some j log the evet j holds ad we have h j = 0, the j, holds. Let us suppose that j holds ad let us coside e j+1. We distiguish seveal possibilities as we aleady uled out the occuece of a edge fom AF j,0 AF j,3 w.h.p., we omit these possibilities: the electoic joual of combiatoics 17 010, #R6 13
1. e j+1 AF j,1, e j+1 S / S j, that is, the oly commo elemet x of e j+1 ad S is ot cotaied i at least edges of F j. Thee ae at most 3 h j 5 3 = Θ 3 possibilities fo such a e j+1, as we ca pick e j+1 S i 3 h j ways ad e j+1 must meet all the edges ot cotaiig x thee ae at least of them outside S whee those edges ae paiwise disjoit. Thus the pobability of pickig such a edge is O1/.. e j+1 AF j,1, e j+1 S S j ad e j+1 does ot ceate a 3-sta. We ca pick the oly elemet x of e j+1 S j i h j ways. We kow that e j+1 must meet the edge of F 3 that does ot cotai x outside S ad to ot ceate a 3-sta e j+1 must meet at least oe of the edges of F 3 cotaiig x outside S as well. These sets ae paiwise disjoit outside S, thus thee ae at most h j 4 5 3 = Θ 3 possibilities agai, thus the pobability of this to happe is O1/. 3. e j+1 AF j,1, e j+1 S S j ad e j+1 ceates a 3-sta, i.e. C j+1,3,1 holds. Numbe of possibilities: agai, we ca pick the oly elemet x of e j+1 S j i h j ways ad e j+1 must meet the edge of F 3 that does ot cotai x outside S. Thus thee ae at most h j 4 possibilities ad if ej+1 does ot meet ay of the edges of F 3 cotaiig x outside S, the a 3-sta is ceated, thus thee ae at least h j 3+3 to choose e j+1. Theefoe the umbe of possibilities is h j 1 + Θ1/. 4. e j+1 AF j,, h j+1 = h j, that is, the oly elemet x of S which is ot i e j+1 does ot belog to S j. To pick x we have 3 h j possibilities ad the fact that the othe elemets of S do belog to e j+1 assues that e j+1 itesects all edges i F j, thus the umbe of possible e j+1 s is 3 h j 1 + Θ1/. 5. e j+1 AF j,, h j+1 = h j 1. The oly diffeece to the pevious case is that ow we have to pick x fom S j ad thus the umbe of possible e j+1 s is h j 3 1+Θ1/. The pobability that the fist two possibilities happe at least oce fo some j log is O log thus eithe 3 happes o the pocess always picks a edge accodig to 4 o 5. The pobability that possibility 4 happes with h j < 3 at least log times while 3 does ot occu at all is O3 log. Thus we obtai that fo some j log eithe possibility 3 happes o we will have h j = 0 which is equivalet to j,. The pobability that if h j > 0, the possibility 5 happes befoe possibility 3 is h 3 j h 3 j + hj 1 + O1/ log = 1 1 + o1. 1 Thus the pobability that j, holds fo some j log is 3 1 1 + o1, 1 the electoic joual of combiatoics 17 010, #R6 14
while the pobability that C j,3,1 holds fo some j log is 1 3 1 1 + o1. 1 Note that Lemma 3.3 implies PC log,3,1 C j,3,1 = 1 o1 i fact, it states that C j,3,1 implies the appeaace of a -double boom, which is much moe tha C log,3,1. Also, P 3R j +1, j, = O1/ fo ay j j log as the umbe of -sets f meetig all edges of F j ad itesectig S i oe elemet is at most 3 5 3 = Θ 3 pick {x} = S f i 3 ways ad f must meet the edges of the -tiagle, assued by j,, that do ot cotai x thus P 3R log, j, = O log. Lemma 4.1 assets that, i the case of costat, F log cotais eithe a 3-sta o a -tiagle w.h.p. By Lemma 3.3, i the fome case F j cotais a -double boom w.h.p. fo sufficietly lage j ad thus by Popositio.3 assues that the pocess esults a HM-type hypegaph. The ext lemma states that i the latte case F log cotais a -tiagle w.h.p. ad thus by Popositio. assues that the pocess outputs a -3 hypegaph. Lemma 4. If 3 is a costat, the P 3R log, log, = 1 o1. Poof. With the otatio of Lemma 4.1, the evet log, implies that fo ay j log we have 9 5 3 M j,0 6 M j,1 M j,3. 6 3 3 Futhemoe, if all edges i F j itesect the base S of the -tiagle cotaied i F log i at least elemets, the 3 3 3 j M j, 3. Thus the pobability, that F log will cotai a edge e with e S is O log. Let β j deote the lagest itege k such that F j cotais a subhypegaph of a - tiagle with k edges, i paticula β j = 3 if ad oly if F j cotais a -tiagle. Let us itoduce the followig adom vaiable { 1 if βj α W j = j+1 o β j = 3 0 othewise. If β j < 3 ad if all edges i F j itesect the base S of the -tiagle cotaied i F log i at least elemets, the P 3R W j = 1 βj 3 3 i=0 M j,i 1/3 o1. the electoic joual of combiatoics 17 010, #R6 15
Theefoe log P 3R β log < 3 log, = P 3R j=log W j < 3 log, log PBilog, 1/3 o1 < 3 + O. Poof of Theoem 1.4. Lemma 3.1 assues that holds w.h.p. Lemma 4.1 ad Lemma 4. togethe with Popositio. ad Popositio.3 poves Theoem 1.4. 5 The Put-Off Model I this sectio, we pove Theoem 1.5. The poof is simila to that of Theoem 1.3 as agai fo a lage eough j we will pove the existece of a -double boom i F j to assue that E HM happes. The mai diffeece betwee the models is that while i the Thid Roud Model the set of semi-keels of the -double boom is aleady detemied afte e 4 is picked i the Put-Off Model thee might be lots of possibilities fo this set eve late. Let us defie j 1 = mi{ 1 1/4, } ad j = mi{ 1 1/, 3 }. Ou fist lemma states that if F 3 is a 3-sta, the w.h.p. the keel of this 3-sta is cotaied i all edges of F j. Lemma 5.1 If = O 1/3, the P PO B j A 3 = 1 o1. Poof. Obseve that if B j holds, the the umbe of sets i AF j cotaiig a elemet of e F j e is at least 1 j. The umbe of -sets cotaiig a fixed elemet x ad meetig a fixed -set f AF j i exactly oe elemet with x / f is 1 ad eve if all of them belog to F j the the othes ae i AF j. Suppose fist that = o 1/4. The by Lemma.6 we may assume that A holds ad thus fo ay j the umbe of sets i AF j ot cotaiig the keel of F j is at most 1. Thus we have P PO B j+1 B j, A 3 1 1 j 1 1 1. If is costat, the multiplyig the last atio by j 1 1/ still gives a boud that teds to 0. If teds to ifiity, the 3 1 1 1 0 holds as = o1/ ǫ. Suppose ow that = ω 1/5 ad thus j = 3. The by Lemma.6 we kow that A 1/5 holds w.h.p. I this case fo ay j with 1/5 j 3 = j the umbe of sets i the electoic joual of combiatoics 17 010, #R6 16
AF j ot cotaiig the keel of F 1/5 is at most 1 1/5 1/5 1 1/5 ad thus we have 1/5 1/5 P PO B j+1 B j, A 3 11/5 1/5 1 1 11/5 1/5 1 j 1 1 1/5 1/5 1 j=0 1/5 1 j 1 j = 1/5 exp O. Theefoe P PO B 3 A 3 3 1/5 exp O + P PO A 1/5 A 3 0. Lemma 5. assets that if the edges of F 3 fom a 3-sta, the F j1 cotais a -sta ad its keel belogs to all but at most oe edge of F j1. Lemma 5. If = O 1/3, the Poof. We coside two cases. Case I = o 1/4 P PO A j 1, A 3 = 1 o1. I this case Lemma.6 shows that P PO A A 3 = 1 o1 ad we ae doe as A = A, ad A, B 3 implies A j, fo ay 3 j. Case II ω 1/5 = = O 1/3 I this case j 1 =. Usig Lemma.6 ad Lemma 5.1 we ca assume that A 1/5 ad B 3 hold. Let x deote the keel of F 1/5. Let us defie T = {t : x / t, t ei i, 1 i 1/5 }. Clealy, we have 1/5 1 T 1 1/5. 1/5 Fo ay t T ad j 1/5 let ν t,j deote the maximum umbe k such that thee exists a k-sta i F j such that t is a tasvesal of the sets of the k-sta. Clealy, the evet A j, holds if ad oly if thee is a t T with ν j,t =. Let us itoduce the followig adom vaiables: { 1 if νj,t ν X j,t = j+1,t o ν j,t = o t / AF j F j 0 othewise. Let us wite futhemoe X t = j= X 1/5 j,t. Obseve that if X t fo all t T, the holds. Ideed, by the defiitio of the Put-Off Pocess thee must always be a A, the electoic joual of combiatoics 17 010, #R6 17
t T which belogs AF j F j ad fo this t the fact X t shows that thee exists a -sta i F 1/3 of which t is a tasvesal. Let us boud fom below the pobability P PO X j,t = 1 fo ay t ad j 1/5. If t / AF j F j ad ν j,t < as othewise X j,t = 1 fo sue, the let us fix a ν j,t -sta B showig this. The keel of B must be x. The umbe of -sets cotaiig x but othewise disjoit fom b B b that meet t i exactly oe elemet is νj,t 1 1 ν j,t. O the othe had, fo ay j 1/5 we have AF j {e : x e} T. Sice T 1 1/5 1/5 1 1 1/5 1, we have AFj 1 1 ad thus 1 P PO X j,t = 1 1 1 1 1 + 3 3 1 Theefoe we have = Ω 1 P PO t T : X t T P = Ω 1 1 = Ω. Bi, Ω expo log exp Ω 0. Lemma 5.3 states that if F j1 cotais a -sta, the F j cotais a -double boom w.h.p. ad thus with Popositio.3 assues that the pocess outputs a HM-type hypegaph. Lemma 5.3 If = O 1/3, the we have P PO A j, B 3, A, = 1 o1. Poof. The poof is simila to that of Case II i the poof of Lemma 5.. Fo ay j j 1 let µ j deote the lagest itege d such that F j cotais a d-patial -double boom. Sice a -sta is a 0-patial -double boom, µ j is well-defied. Let us itoduce the followig adom vaiables: { 1 if µj µ Y j = j+1 o µ j = 0 othewise. Let us wite futhemoe Y = j j=j 1 Y j. Let us shape the uppe boud o AF j fom Lemma 5. by cosideig a µ j -patial -double boom B with keel x. Evey set f i AF j must meet b B b\{x} as othewise {f} F j would cotai a +1-sta which is impossible to exted to a o-tivially itesectig hypegaph. Thus the umbe of sets the electoic joual of combiatoics 17 010, #R6 18
i AF j cotaiig x is at most. The umbe of sets i AFj ot cotaiig x is at most 1 as A j, holds. Thus AF j + 1 3. O the othe had, by Popositio.3 evey t AF j with x / t cotais all the semikeels of B ad meets all the loely figes of B. Note that thee is at least oe such set by the defiitio of the Put-Off Pocess, say t j. Obseve that ay -set e cotaiig x with t j e = 1, e b B b = ad t j e belogig to a loely fige of B is i AF j, futhemoe if such a set is chose to be e j+1 by the Put-Off pocess, the µ j+1 = µ j + 1. Thus we obtai P PO Y j = 1 µ j 1 1 3 1 3 as = O 1/3. Now we have 1 P PO A j, B j, A j 1, P PO Y < B j, A j 1, P 1 Ω 1 1 = Ω, 1 Bi j j 1, Ω < 0, as if is costat, the 1/ is costat ad j j 1, while if = ω1, the j j 1 = 3. Poof of Theoem 1.5. Let c = < /. If c teds to 0, the Lemma.5, Lemma 5.1, Lemma 5. ad Lemma 5.3 assues that fo a suitably chose j F j cotais a -double-boom w.h.p. ad thus, by Popositio.3, E HM holds w.h.p. If c teds to c, the Lemma.5 states that the pobability that A 3 holds teds to 1 i which case, just as fo = ω 1/3, E 1+c 3 HM holds. Lemma.7 assues that B 3 holds with pobability tedig to 1 1. As if B 1+c 3 3 holds, the the Thid Roud Model ad the Put-Off Model coicide ad thus by Theoem 1.3 we obtai that P PO E HM B 3 = 1 P 3R E HM 1 + o1 teds to. Hece P 1 1+c 3 /3 POE HM teds to + 1 1 1 as 1+c 3 1+c 3 1+c 3 /3 claimed i the theoem. If c teds to ifiity, the ote that fo ω 1/3 = = o /3 a cosequece of Lemma.8 is that w.h.p the Thid Roud Model ad the Put-Off Model ae the same, while Lemma 3.5 poves Theoem 1.5 fo ω 1/ = < /. Ackowledgemet. I would like to thak the aoymous efeee fo his/he caeful eadig ad valuable commets. Refeeces [1] N. Alo; J. Spece, The Pobabilistic Method, Wiley-Itesciece Seies i Discete Mathematics ad Optimizatio. Wiley-Itesciece, New Yok, secod editio, 000. the electoic joual of combiatoics 17 010, #R6 19
[] J. Balogh; T. Bohma; D. Mubayi, Edos-Ko-Rado i Radom Hypegaphs, to appea i Combiatoics, Pobability ad Computig. [3] T. Bohma; C. Coope; A. Fieze; R. Mati; M. Ruszikó, O adomly geeated itesectig hypegaphs, The Electoic Joual of Combiatoics, 10 003 R 9. [4] T. Bohma; A. Fieze; R. Mati; M. Ruszikó; C. Smyth, Radomly geeated itesectig hypegaphs. II. Radom Stuctues Algoithms 30 007, 17 34. [5] P. Edős; C. Ko; R. Rado, Itesectio theoems fo systems of fiite sets. Quat. J. Math. Oxfod Se. 1 1961 313 30. [6] A.J.W. Hilto; E.C. Mile, Some itesectio theoems fo systems of fiite sets. Quat. J. Math. Oxfod Se. 18 1967 369 384. the electoic joual of combiatoics 17 010, #R6 0