Hypergraph Independent Sets

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1 Uivesity of Nebasa - Licol DigitalCommos@Uivesity of Nebasa - Licol Faculty Publicatios, Depatmet of Mathematics Mathematics, Depatmet of 2013 Hypegaph Idepedet Sets Joatha Cutle Motclai State Uivesity, oatha.cutle@motclai.edu A. J. Radcliffe Uivesity of Nebasa-Licol, amie.adcliffe@ul.edu Follow this ad additioal wos at: Cutle, Joatha ad Radcliffe, A. J., "Hypegaph Idepedet Sets" Faculty Publicatios, Depatmet of Mathematics This Aticle is bought to you fo fee ad ope access by the Mathematics, Depatmet of at DigitalCommos@Uivesity of Nebasa - Licol. It has bee accepted fo iclusio i Faculty Publicatios, Depatmet of Mathematics by a authoized admiistato of DigitalCommos@Uivesity of Nebasa - Licol.

2 Combiatoics, Pobability ad Computig , c Cambidge Uivesity Pess 2012 doi: /s Hypegaph Idepedet Sets JONATHAN CUTLER 1 ad A. J. RADCLIFFE 2 1 Depatmet of Mathematical Scieces, Motclai State Uivesity, Motclai, NJ 07043, USA oatha.cutle@motclai.edu 2 Depatmet of Mathematics, Uivesity of Nebasa Licol, Licol, NE , USA aadcliffe1@math.ul.edu Received 17 Apil 2012; evised 8 August 2012; fist published olie 11 Octobe 2012 The study of extemal poblems elated to idepedet sets i hypegaphs is a poblem that has geeated much iteest. Thee ae a vaiety of types of idepedet sets i hypegaphs depedig o the umbe of vetices fom a idepedet set allowed i a edge. We say that a subset of vetices is -idepedet if its itesectio with ay edge has size stictly less tha. The Kusal Katoa theoem implies that i a -uifom hypegaph with a fixed size ad ode, the hypegaph with the most -idepedet sets is the lexicogaphic hypegaph. I this pape, we use a hypegaph egulaity lemma, alog with a techique developed by Loh, Pihuo ad Sudaov, to give a asymptotically best possible uppe boud o the umbe of -idepedet sets i a -uifom hypegaph. AMS 2010 Mathematics subect classificatio: Pimay 05C65 Secoday 05D05, 05C69 1. Itoductio The study of idepedet sets i gaphs has a log histoy. Recetly, extemal poblems elated to maximizig the umbe of idepedet sets i gaphs have bee a active aea of eseach. Kah [7] detemied which egula bipatite gaphs with vetices have the most idepedet sets, ad his theoem was ecetly exteded to all egula gaphs by Zhao [15]. It is a cosequece of the Kusal Katoa theoem [9, 8] that the lexicogaphic gaph has the lagest umbe of idepedet sets amog gaphs of fixed ode ad size see, e.g., [2]. Idepedet sets i hypegaphs have also bee well studied. Much of this eseach has focused o detemiig algoithms fo fidig idepedet sets i -uifom hypegaphs see, e.g., [14]. I this pape we detemie asymptotically the maximum umbe of idepedet sets possible fo a -uifom hypegaph o vetices ad m edges. Idepedet sets i hypegaphs ae a bit moe complicated tha those i gaphs sice they ca be defied i a umbe of ways depedig o how may vetices fom a idepedet set ae allowed i a edge.

3 10 J. Cutle ad A. J. Radcliffe Fo us, a hypegaph H is a odeed pai V,E, whee V is a fiite set ad E PV. We say a hypegaph H =V,E is -uifom if E V, whee V = {E V : E = }. Foa hypegaph H =V,E, we wite H fo V ad eh fo E. Defiitio. Fo a -uifom hypegaph H =V,E ad a itege with 1, wesay that a set I V H is -idepedet if I E <fo all E E.LetI H be the collectio of all -idepedet subsets of V ad i H = I H. Example 1. If G is a gaph, i.e., a2-uifom hypegaph, the I 2 G is the collectio of all idepedet sets i the gaph G. O the othe had, I 1 G is simply the collectio of subsets disoit fom all edges of G. Thus, i 1 G =2 0, whee 0 is the umbe of isolated vetices i G. I the case =, the maximum umbe of idepedet sets possible i a -uifom hypegaph is ow exactly as a cosequece of the Kusal Katoa theoem. Fo the sae of completeess, we iclude this esult ad its poof. Theoem 1.1. If H =V,E is a -uifom hypegaph with H = ad eh =m, the i H i L,m, whee L,m is the -gaph o [] ={1, 2,...,} whose edges ae the fist m elemets i the lexicogaphic odeig 1 o []. Poof. Let us wite I uppe shadow of H o level,theset { V H = B Thus H fo I H V i H = =0. Note that I I H if ad oly if I is ot i the } : E a edge of H such that E B. I H 1 = + =0 1 + =0 = i L,m, = = H L,m whee the iequality H L,m is the cotet of the Kusal Katoa theoem. Maximizig the umbe of 1-idepedet sets is tivial, sice fo a -uifom hypegaph H with vetices, i 1 H 2, 1 Fo A, B [], wesaya< lex B if ad oly if mia B A.

4 Hypegaph Idepedet Sets 11 whee is the uique itege fo which 1 <eh. Just as i Example 1 fo gaphs, 1-idepedet sets i a hypegaph H ae simply subsets of the isolated vetices of H-vetices ot i ay edge of H. This boud is achieved by ay hypegaph whose edges ae cotaied i some set of size. Fo 1 << the poblem of maximizig the umbe of -idepedet sets i -uifom hypegaphs is ope. If we follow the appoach of Theoem 1.1 ad thi about I H fo some <,wehave whee the lowe shadow H = { B I H = V H, } V : E a edge of H such that B E. The cofiguatio of sets that miimizes the lowe shadow, i.e., the colex hypegaph, is vey diffeet tha that which miimizes the uppe shadow, i.e., the lex hypegaph. It is thus vey difficult to get exact esults givig uppe bouds o the umbe of idepedet sets i hypegaphs. We give a asymptotically best possible boud o i H i tems of the umbe of idepedet sets i a split hypegaph, which we defie below. Ou appoach follows that of Loh, Pihuo ad Sudaov i [10], whee they detemie asymptotically the maximum umbe of q-colouigs of a gaph G with vetices ad m edges. They use Szemeédi s Regulaity Lemma [12] i a cleve way. Give a egula patitio of the vetex set ad ad q-colouig of G, they associate a colouig of the auxiliay gaph. Sice the egula patitio has a bouded umbe of pats, thee ae oly a costat umbe of possible auxiliay colouigs ad oe eed oly coside, asymptotically, those colouigs of G coespodig to a fixed auxiliay colouig. This allows them to get good cotol o the poblem of maximizig the umbe of q-colouigs. We adapt this appoach to pove a asymptotic boud o the umbe of -idepedet sets i a -uifom hypegaph. We pove that the way to get may -idepedet sets is, asymptotically, to have a vey lage -idepedet set, all of whose subsets ae, of couse, -idepedet. Aothe way to say this is that the asymptotic extemal gaphs will be what we call split hypegaphs, which ae aalogous to split gaphs. I adaptig the appoach of Loh, Pihuo ad Sudaov we eed to use a hypegaph egulaity lemma. I Sectio 2, we discuss both the hypegaph egulaity lemma ad pelimiay esults about split hypegaphs, defied below. Defiitio. uifom hypegaph with vetex set V = A B ad edge set { V E : E A < The -split -gaph with patitio A, B, deoted S A, B, is defied as the - Whe we ae ot coceed with the idetity of the sets A ad B, we wite S, fo a -split -gaph with A = ad B =. Welet See Figue 1 fo a example. }. s, =e S,.

5 12 J. Cutle ad A. J. Radcliffe A B Figue 1. Colou olie The types of edges i the split 3-gaph S 3 2 A, B. We ca ow state the mai esult of the pape. I the statemet, ad fo the est of the pape, all o1 temsgotozeoas goes to ifiity. Mai Theoem. Let ad 1be positive iteges with 1. Give η>0 ad ay -uifom hypegaph H o vetices with η eh 1 η,ifwelet be maximal such that the eh s,, i H 2 1+o1. I Sectio 2, i additio to povig some pelimiay lemmas, we will show that this esult is asymptotically best possible by showig that i S, =1+o12, fo / bouded away fom 0 ad 1. I Sectio 3, we pove the Mai Theoem. 2. Pelimiaies 2.1. Hypegaph egulaity Thee ae ow highly sophisticated hypegaph egulaity lemmas available: see, e.g., [1], [5], [6], [11] ad [13]. We, howeve, eed oly a simple vesio which ca be foud i [3] o [4], fo example. I ode to state the egulaity lemma, we eed fist to defie a ε-egula patitio, the stuctue guaateed by the egulaity lemma. I what follows, we use the stadad otatio that if H =V,E is a -uifom hypegaph ad W 1,W 2,...,W ae disoit subsets of the vetex set, the H[W 1,W 2,...,W ]={E E : E W i =1, i []}. Defiitio. Let H =V,E be a -uifom hypegaph. Give ε>0, we say a -tuple W 1,W 2,...,W of disoit subsets of V is ε-egula if, fo all sequeces S i 1 of subsets

6 Hypegaph Idepedet Sets 13 S i W i with S i ε W i fo all i [],wehave eh[w 1,W 2,...,W ] 1 W eh[s 1,S 2,...,S ] i 1 S i <ε. We say a patitio {V 1,...,V t } is a ε-egula patitio of H if 1 V 1 V 2... V t V 1 +1, ad 2 the -tuple V i1,v i2,...,v i is ε-egula fo all but ε t of the -sets {i1,i 2,...,i } i [t]. The followig hypegaph egulaity lemma ca be ead out of, fo example, a esult of Czygiow ad Rödl [3]. Theoem 2.1. Fo all, m ad ε>0, thee exists M,L such that give ay -uifom hypegaph H =V,E with V L, thee is a ε-egula patitio {V 1,...,V t } of H with m t M Split hypegaphs We ow peset some chaacteistics of the -split -gaphs, S,, which wee defied i Sectio 1. To develop ituitio, we biefly discuss the =2case. Example 2. Note that i the case whe =2, oly two values fo ae of iteest, amely 1 ad 2. If 3, the S 2, is a complete gaph. I the case whe =1, we have that S 2 1, is the disoit uio of the empty gaph E ad the complete gaph K. Whe =2, similaly, S 2 2, is the oi2 of E ad K. We ow pove a sequece of lemmas coceig hypegaphs which we will eed fo the poof of the Mai Theoem. The fist lemma gives the umbe of edges ad -idepedet sets i the split -gaphs. Fo coveiece, we wite = i=0. i Fom this poit fowad, we will fix a itege ad discuss -uifom hypegaphs. Also, we will fix a itege with 1. Ou aim is to pove the Mai Theoem fo these values of ad, which will appea i ou lemmas without futhe commet. Lemma 2.2. Let be a positive itege with.if A = ad B =, the the umbe of edges i S = S A, B is 1 es = i=0 i i. 2 The oi of gaphs G ad H is the gaph with vetex set the disoit uio of V G ad V H ad edge set EG EH {xy : x V G,y V H}.

7 14 J. Cutle ad A. J. Radcliffe If + 1, i.e., S is eithe complete o empty, the the umbe of -idepedet sets i S is i S = Poof. The fist calculatio is staightfowad. Fo the secod, it is easy to chec that i the elevat age of, aysetof vetices ot cotaied i A is cotaied i a edge. Coollay 2.3. povided ξ</<1 ξ. Give 0 <ξ<1/2, we have i S, =1+o12, The ext techical lemma bouds the diffeece i the umbe of edges betwee split gaphs with adacet values of. We eed to show that the diffeece is lage to esue that chagig by a costat multiple of chages the popotio of edges i the hypegaph by a costat amout. Lemma 2.4. Give 0 <ξ<1/2 thee exists ζ>0 such that wheeve is a itege with 2max, /ξ ad [] satisfies ξ,1 ξ, we have s 1, +1 s, +ζ 1. Poof. ad Witig S = S 1, +1ad S = S,, we see that S cotais S 1 es es = 1 1 1! 1 1!! +! 1 ξ ξ 2 2 ξ 1 = 2 1 1!! 1. We ca clealy set ζ = ξ 1 / 2 1 1!!. S Ou last lemma discusses the elatioship betwee the umbe of edges i the split hypegaph, ad the atio /. We mae the followig defiitio. Defiitio. Give a itege ad e with 0 e, we wite, e = max{ : s, e},

8 Hypegaph Idepedet Sets 15 fo the lagest such that the split gaph S, has at least e edges. This is clealy welldefied sice s 0,= ad s, 0 = 0. Note futhe that s, is a deceasig fuctio of fo fixed. We pove that if e/ is bouded away fom 0 ad 1 the so is, e/. Lemma 2.5. Give ν 0, 1/2, thee exists ξ = ξν 0, 1/2 such that fo sufficietly lage as a fuctio of ν, we have the followig: if e satisfies ν <e<1 ν, the =, e satisfies ξ< < 1 ξ. Poof. Fo the lowe boud by Lemma 2.2, we wat to show that thee is a ξ such that 1 s ξ, ξ = i=0 ξ i ξ i 1 ν. Pic ξ 1 with 0 <ξ 1 < 1 1 ν 1/. To show the above with ξ = ξ 1, we boud the sum by the i =0tem. Thus, 1 s ξ1 ξ1 ξ 1, ξ 1 = i i i=0 1 ξ1 =1 o11 ξ 1 > 1 ν, fo sufficietly lage. I the othe diectio pic ξ 2 with 0 <ξ 2 <ν/ /2.Weleti be the value of i with 1 i 1 that maximizes 1 ξ 2 ξ2 i i. Note that i 1 <.Now, 1 s ξ2 ξ2 ξ 2, ξ 2 = i i i=0 1 ξ2 ξ2 i i <1 ξ 2 i ξ2 i i i.

9 16 J. Cutle ad A. J. Radcliffe Sice we wat a uppe boud, we ca eglect the factos of 1 ξ 2 ; sice i <, thee is at least oe facto of ξ 2. So, s ξ 2, ξ 2 ξ 2 i i i = ξ o1 i i = ξ o1 i ξ o1 /2 <ν, fo sufficietly lage. Lettig ξ = miξ 1,ξ 2, the lemma follows. 3. Poof of Mai Theoem We estate the Mai Theoem befoe pesetig its poof. Mai Theoem. Let ad 1be positive iteges with 1. Give η>0 ad ay -uifom hypegaph H o vetices with η eh 1 η,ifwelet be maximal such that the eh s,, i H 2 1+o1. Poof. Give ɛ>0, we wat to show that fo sufficietly lage ad H a -uifom hypegaph with η eh 1 η,wehave i H 2 1+ɛ,e, 3.1 whee we have witte e fo eh. We will poceed by itoducig a paamete δ which goves ou use of the egulaity lemma; at the ed of the poof we will choose δ sufficietly small so as to achieve the boud i 3.1. Ou poof poceeds i a sequece of steps. Step 1. Give 0 <δ<1, thee exists a cleaed-up subhypegaph H of H ad a δ-egula patitio {V 1,V 2,...,V t } of H such that a eh eh δ, b all edges of H spa pats of the patitio, ad c all subgaphs H [V i1,v i2,...,v i ] ae eithe empty o δ-egula with desity at least δ.

10 Hypegaph Idepedet Sets 17 To show this, fist apply Theoem 2.1 with ε = δ/4 ad m sufficietly lage that 1 1 i > 1 δ m 2 i=1 to get a suitable patitio {V 1,V 2,...,V t }. We get H by fist deletig all edges that do ot spa pats of the patitio. Taig ito accout the fact that the sizes of the V i ae each withi oe of /t, ad the umbe of pats i the patitio is bouded by M, the umbe of -sets i V H that do ot spa pats of the patitio is at most [ 1 + o1 1 i=1 1 i t ] =1+o1 [1 1 1 i ]! t i=1 [ o1 1 1 i ] m i=1 < 1 + o1 δ o1 δ 4 < 3δ 8 fo sufficietly lage. Cetaily, the, the umbe of edges of H that do ot spa pats of the patitio is at most 3δ 8.Now,if{i 1,i 2,...,i } has the popety that H[V i1,v i2,...,v i ] eithe has desity less tha δ o is ot δ-egula, we delete all edges of H[V i1,v i2,...,v i ]. The total umbe of such deleted edges is at most t δ + δ t < 5δ t 4 t 8. Recall that the patitio was chose to be δ/4-egula. The hypegaph H is the esult of deletig all of the above edges fom H. Step 2. Now we focus ou attetio o -idepedet sets with the popety that if they itesect a pat V i, the they have easoably lage itesectio with that pat. Defiitio. Give a -idepedet set I i H, we defie a subset DI [t] by DI ={i [t] : I V i δ V i }. We call a -idepedet set obust if I V i = fo all i/ DI. WeletR be the set of obust -idepedet sets i H. Also, fo coveiece we defie V D,foD [t],tobe V D = i D V i. The followig lemma poves that a easoable popotio of -idepedet sets ae obust.

11 18 J. Cutle ad A. J. Radcliffe Lemma 3.1. Thee exists c δ > 0 such that ad c δ teds to zeo as δ teds to zeo. Poof. Defie a map f : I H R by R exp c δ i H fi =I V DI. We show that each I 0 R has at most exp2δ δ l δ pe-images ude f. Fist ote that I \ fi δ; theefoe the umbe of I such that fi =I 0 is at most, usig Stilig s fomula to boud the lagest tem, δ δ δexpδ s δ s=0 δ 1 expδexpδ =exp2δ δl δ. δ Settig c δ =2δ δl δ poves the lemma. Step 3. We have show that thee ae ot too may fewe obust -idepedet sets tha - idepedet sets. Now we show that the existece of may obust -idepedet sets costais H to be a subgaph of a -split hypegaph. To be pecise, fo D [t], welets D be the -split hypegaph S V D,VD c, ad show the followig. Lemma 3.2. If I R ad DI =D, the H S D, o, i othe wods, V D is -idepedet. Poof. We eed to show that if {V i1,v i2,...,v i } is a set of blocs with {i 1,i 2,...,i } D, the H [V i1,v i2,...,v i ] is empty. If it is ot empty the it is δ-egula. Fo i {i 1,i 2,...,i } D, wehave I V i δ V i, so, by the δ-egulaity, thee is a edge E of H with E V i I V i fo all i {i 1,i 2,...,i } D. I paticula, E I, cotadictig the -idepedece of I. Step 4. Fially, we pove 3.1, that log 2 i H 1 + ɛ, e. Fo D [t], weletr D be the collectio of obust -idepedet sets suppoted o D, i.e., R D = {I R : DI =D}.FixaD with R D maximal, so that We estimate as follows: R D 2 t R 2 M R. i H i H expc δ R

12 Hypegaph Idepedet Sets 19 expc δ 2 M R D expc δ 2 M 2 V D. The fist iequality follows fom the fact that H H, the secod fom Lemma 3.1, ad the thid fom. The fial estimate is simply the fact that if I is a obust idepedet set with DI =D, the I V D. Taig logaithms, we have log 2 i H c δ l 2 + V D + M. 3.2 We ow eed to boud V D. Lemma 3.3. Thee exists c δ > 0 such that ad c δ teds to zeo as δ teds to zeo. V D, e+c δ Poof. Let e = eh ad =, e = max{ : s, e }. Sice H is a subgaph of S D, by Lemma 3.2, we have V D. We apply Lemma 2.5 with ν = η/2 to get a suitable ξ 0, 1/2. Recall that e >η δ. So, fo sufficietly lage ad δ<η/4!, wehave η < η 2!δ e eh 1 η < 1 η. 2 2 So, Lemma 2.5 implies that ξ<, e < 1 ξ. Thus, by Lemma 2.4, thee exists ζ>0such that fo all with, e,wehave I paticula, s 1, +1 s, +ζ 1. δ e e Theefoe,, e δ ζ We set c δ = 2δ ζ s, e+1,, e 1 s,, e 1ζ 1. 2δ +1 ζ.wehave V D, e+ 2δ ζ. ad the lemma is poved, otig that ζ does ot deped o δ.

13 20 J. Cutle ad A. J. Radcliffe Applyig Lemma 3.3 to 3.2, we have, fo sufficietly lage, log 2 i H c δ l 2 +, e+c δ + M, e+ 1 + ɛ, e. cδ l 2 + c δ ξ, e+m The peultimate step follows fom Lemma 2.5. The fial boud comes fom choosig δ sufficietly small. Refeeces [1] Chug, F. R. K Regulaity lemmas fo hypegaphs ad quasi-adomess. Radom Stuct. Alg [2] Cutle, J. ad Radcliffe, A. J Extemal poblems fo idepedet set eumeatio. Electo. J. Combi. 18 #R169. [3] Czygiow, A. ad Rödl, V A algoithmic egulaity lemma fo hypegaphs. SIAM J. Comput [4] Fieze, A. ad Kaa, R Quic appoximatio to matices ad applicatios. Combiatoica [5] Gowes, W. T Quasiadomess, coutig ad egulaity fo 3-uifom hypegaphs. Combi. Pobab. Comput [6] Gowes, W. T Hypegaph egulaity ad the multidimesioal Szemeédi theoem. A. of Math [7] Kah, J A etopy appoach to the had-coe model o bipatite gaphs. Combi. Pobab. Comput [8] Katoa, G A theoem of fiite sets. I Theoy of Gaphs: Poc. Colloq., Tihay, 1966, Academic Pess, pp [9] Kusal, J. B The umbe of simplices i a complex. I Mathematical Optimizatio Techiques R. Bellma, ed., Uivesity of Califoia Pess, pp [10] Loh, P.-S., Pihuo, O. ad Sudaov, B Maximizig the umbe of q-coloigs. Poc. Lodo Math. Soc [11] Rödl, V., Nagle, B., Soa, J., Schacht, M. ad Kohayaawa, Y The hypegaph egulaity method ad its applicatios. Poc. Natl. Acad. Sci. USA [12] Szemeédi, E Regula patitios of gaphs. I Poblèmes Combiatoies et Théoie des Gaphes, Vol. 260 of Colloq. Iteat. CNRS, CNRS, pp [13] Tao, T A vaiat of the hypegaph emoval lemma. J. Combi. Theoy Se. A [14] Yuste, R Fidig ad coutig cliques ad idepedet sets i -uifom hypegaphs. Ifom. Pocess. Lett [15] Zhao, Y The umbe of idepedet sets i a egula gaph. Combi. Pobab. Comput