New lowe bounds fo the independence numbe of spase gaphs and hypegaphs Kunal Dutta, Dhuv Mubayi, and C.R. Subamanian May 23, 202 Abstact We obtain new lowe bounds fo the independence numbe of K -fee gaphs and linea k- unifom hypegaphs in tems of the degee sequence. This answes some old questions aised by Cao and Tuza [7]. Ou poof technique is an extension of a method of Cao and Wei [6, 20], and we also give a new shot poof of the main esult of [7] using this appoach. As bypoducts, we also obtain some non-tivial identities involving binomial coefficients, which may be of independent inteest. Intoduction Fo k 2, a k-unifom hypegaph H is a pai V H, EH whee E V H k. A set I V H is an independent set of H if e I fo evey e EH, o equivalently, I k EH =. The independence numbe of H, denoted by αh, is the maximum size of an independent set in H. Fo u V H, its degee in H, denoted by d H u, is defined to be {e EH : u e} we omit the subscipt if it is obvious fom the context. Thoughout this pape, we use t to denote k except in some places whee it stands fo some eal value the coect meaning can be easily infeed fom the context. Also, we use the tem gaph wheneve k happens to be 2. A k-unifom hypegaph is linea if it has no 2-cycles whee a 2-cycle is a set of 2 hypeedges containing at most 2t vetices. The dual of the above definition says that a linea hypegaph is one in which evey pai of vetices is contained in at most one hypeedge. In [9], Tuán poved a theoem giving a tight bound on the maximum numbe of edges that a K -fee gaph can have, which has since become the conestone theoem of extemal gaph theoy. Tuán s theoem, when applied to the complement G of a gaph G, yields a lowe bound αg n d+ whee d denotes the aveage degee in G of its vetices. The Institute of Mathematical Sciences, Taamani, Chennai - 6003, India, email: kdutta@imsc.es.in Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois, Chicago, IL-60607, USA email: mubayi@math.uic.edu. Reseach suppoted in pat by NSF gant DMS 0969092 The Institute of Mathematical Sciences, Taamani, Chennai - 6003, India, email: cs@imsc.es.in
Cao [6] and Wei [20] independently poved that αg n v dv+ which is at least d+. The pobabilistic poof of thei esult late appeaed in Alon and Spence s book [5]. One natual n extension of Tuán s theoem to k-unifom hypegaphs H is the bound αh > c k, and d /t this was shown via an easy pobabilistic agument by Spence [5]. Cao and Tuza [7] impoved this bound fo abitay k-unifom hypegaphs. In ode to state thei lowe bound, we need the following definition of factional binomial coefficients fom [0]. Definition Fo t > 0, a 0, d N d + /t td + td +...td a + + := a a!t a What Cao and Tuza [7] showed was that αh v V H dv+/t dv Indeed, an easy consequence of is the following esult.. Theoem. Cao-Tuza [7] Fo evey k 3, thee exists d k > 0 such that evey k-unifom hypegaph H has αh d k dv +. /t v V H As a coollay, one infes the bound of Spence above. Late, Thiele [8] povided a lowe bound on the independence numbe of non-unifom hypegaphs, based on the degee ank a genealization of degee sequence. In this pape, we pove new lowe bounds fo the independence numbe of locally spase gaphs and linea k-unifom hypegaphs. The stating point of ou appoach is the pobabilistic poof of Boppana-Cao-Wei. This appoach, togethe with some additional simple ideas, quickly yields a new shot poof of Theoem. see Section 2 fo the detailed poof.. K -fee gaphs Fo cetain classes of spase gaphs, impovements of the Cao-Wei bound in tems of aveage degee d ae known. Ajtai, Komlós and Szemeédi [2] poved a lowe bound of Ω fo n log d d the independence numbe of tiangle-fee gaphs. An elegant and simple poof was late given by Sheae [2], who also impoved the constant involved. Ajtai, Edős, Komlos and Szemeédi [] showed that fo K -fee gaphs > 3, the independence numbe is lowe-bounded by log d c n/d log, whee c R + depends only on. They also conjectued that the optimal n log d bound is c d. Sheae [4] impoved thei bound to Ω n log d d log log d. Accoding to R. Bopanna [9], the pobabilistic agument in [5] was obtained by him, although it is possible that it was known ealie. 2
Cao and Tuza [7] aised the following question in thei 99 pape : i Can the lowe bounds of Ajtai et al [2] and Sheae [2], [4] be genealized in tems of degee sequences? We answe this question via the following two theoems. Theoem.2 Fo evey ɛ 0, thee exists c > 0 such that the following holds: Evey tiangle-fee gaph G with aveage degee D has independence numbe at least clog D v V G max {D ɛ, dv}. Theoem.3 Fo evey ɛ 0, and 4, thee exists c > 0 such that the following holds: Evey K -fee gaph G with aveage degee D has independence numbe at least c log D log log D v V G max {D ɛ, dv}. A simila bound to Theoem.2 was obtained by Sheae [3], who showed that fo a tianglefee gaph G, αg o log dv v V G dv. We emak that up to multiplicative constants, the function log D v dv is lage than the function log dv v dv used in Sheae s bound..2 Linea Hypegaphs As mentioned ealie, a lowe bound of Ω n/d /t fo an n vetex k-unifom hypegaph with aveage degee d can be infeed fom Theoem.. Cao and Tuza [7] also aised the following question: ii How can one extend the lowe bounds of Ajtai et al [2] and Sheae [2], [4] to hypegaphs? As it tuns out, such extensions wee known fo the class of linea k-unifom hypegaphs. Indeed, the lowe bound /t log d αh = Ω n, 2 d whee H is a linea k-unifom hypegaph with aveage degee d was poved by Duke-Lefmann- Rödl [8], using the esults of [4]. Ou final esult genealizes 2 in tems of the degee sequence of the hypegaph. Theoem.4 Fo evey k 3 and ɛ 0,, thee exists c > 0 such that the following holds: Evey linea k-unifom hypegaph H with aveage degee D has independence numbe at least clog D /t v V H max {D ɛ/t, dv /t }. We also descibe an infinite family of k-unifom linea hypegaphs to illustate that the atio between the bounds of Theoem.4 and 2 can be unbounded in tems of the numbe of vetices. 3
The emainde of this pape is oganized as follows. In Section 2, we give a new shot poof of Theoem.. In Section 3, we apply the analysis in Section 2 to the special case of linea hypegaphs, and obtain a wam-up esult - Theoem 3., which will be helpful in poving the main technical esult, Theoem 4., poved in Section 4. The expession obtained in Theoem 4. plays a cucial ole in the poofs of Theoems.2,.3 and.4; these ae povided in Section 5. In Section 6, we give infinite families of K -fee gaphs and k-unifom linea hypegaphs which illustate that the bounds in Theoems.2,.3 and.4 can be bigge than the coesponding bounds in [2, 4, 8, 2, 4] by abitaily lage multiplicative factos. Finally, in section 8, we state seveal combinatoial identities which follow as simple coollaies of Theoem 4.. 2 A new poof of Theoem. In this section we obtain a new shot poof of Theoem.. theoem which is late used to pove Theoem.. Fist we obtain the following Theoem 2. Fo evey k 2, thee exists a constant c = c k such that any k-unifom hypegaph H on n vetices and m hypeedges satisfies J V H n > c...... A m /k n J whee we sum ove all independent sets J. Poof Let t k n, m denote the LHS of A. Conside any edge e EH. The edge e can belong to at most n k j k non-independent sets of size j. Since thee ae m edges thee ae at most m n k j k sets of size j that ae not independent. Thus, at least n j m n k j k sets of size j ae independent. Hence we have n n k n j k t k n, m m n = m j j k n k > j= n/2m /k j= n 2 2m /k m jk n k j= c k n m /k n/2m /k j= m 2m fo some suitably chosen c k which is close to 2 k+/k. Let H = V, E be a k-unifom hypegaph. Fo k 3 and fo u V with d H u, the link gaph associated with u in H is the t-unifom hypegaph L u = U, F whee U := {v u : e E : {u, v} e} and F = {e\u : u e E}. Let IH denote the collection of independent sets of H. Poof of Theoem.. As mentioned in the Intoduction, the poof is an extension of the technique used in Alon and Spence s book [5]. Let H = V, E be an abitay k-unifom hypegaph. Choose unifomly at andom a total odeing < on V. Define an edge e E to be backwad fo a vetex v e if u < v fo evey u e \ {v}. Define a andom subset I to be 4
the set of those vetices v such that no edge e incident at v is backwad fo v with espect to <. Clealy, I is independent in H. We have E[ I ] = v P v I. If d v = 0, then v I with pobability. Hence, we assume that dv. Fom the definition of I, it follows that v I if and only if fo evey e incident at v, e \ {v} S v = {u V L v : u < v}. In othe wods, S v is an independent set in L v. Let l v = V L v. Then P [v I] = J IL v J!l v J! l v +! = l v + J IL v Applying Theoem 2. to the t-unifom link gaph L v with c = c k, we get c l v cl v P [v I]. l v + dv /k l v + dv + /k Since l v k, we get P [v I] k c/k. By choosing d dv+ /k k = k c/k, we get the lowe bound of the theoem. lv J 3 Lineaity : Pobability of having no backwad edges In this section, we state and pove a wam-up esult on the pobability of having no backwad edges incident at a vetex fo a andomly chosen linea odeing Theoem 3. below. The poblem is the same as in the pevious section, only, now the hypegaph unde consideation is assumed to be linea and we get an explicit closed-fom expession fo this pobability. This esult will be helpful fo the poof of the main technical theoem, given in the next section. Theoem 3. Let H be a linea k-unifom hypegaph and let v be an abitay vetex having degee d. Fo a unifomly chosen total odeing < on V, the pobability P v 0 that v has no backwad edge incident at it, is given by P v 0 = d+/t d Remak. It is inteesting to note that the above expession when summed ove all vetices, is the same bound which Cao and Tuza obtain in [7] using vey diffeent methods, although thei bound holds fo independent sets in geneal k-unifom hypegaphs. We pove the theoem using the well-known Pinciple of Inclusion and Exclusion PIE. Fist we state an identity involving binomial coefficients. Lemma 3.2 Given non-negative integes d and t, d d t + = d+/t d Fo poof see [0], Equation 5.4. Altenatively, it can be poved using the Chu-Vandemonde identity see e.g. [0], Equation 5.93. 5
Poof of Theoem 3. Fistly, obseve that since H is linea, the numbe of vetices that ae neighbos of v is exactly k d = td. Next, notice that since the andom odeing is unifomly chosen, only the elative aangement of these td neighbos and the vetex v, i.e. td + vetices in all, will detemine the equied pobability. Hence the total numbe of odeings unde consideation is td +!. Label the hypeedges incident at v with,..., d abitaily. Fo a pemutation π, we say that π has the popety T S if the edges with labels in S, S [d] ae backwad. Also, say π has the popety T =S if the edges with labels in S ae backwad and no othe edges ae backwad. Fo a set S of hypeedges incident at v, let NT S denote the numbe of odeings having the popety T S, that is, the numbe of pemutations such that the hypeedges in S will all be backwad edges. NT =S is similaly defined. NT S is detemined as follows : Suppose S has hypeedges incident at v. Fo a fixed aangement of the vetices belonging to edges in S, the numbe of pemutations of the emaining vetices is td +!/t +!. In each allowed pemutation, the vetex v must occu only afte the vetices of S i.e. the ightmost position. Howeve the emaining t vetices can be aanged among themselves in t! ways. Thus we have t! NT S = td +! t +! = td +! t +. Clealy, if a pemutation has the popety T S, it has the popety T =S fo some S S. Hence fo evey S [d], NT S = NT =S. Theefoe, by PIE see [6], Chapte 2, S S NT = = S S NT S. NT S = S = d NT [] = Hence we get the equied pobability to be d d td +! P v 0 = t + d d = t +. d td +! t +. td +! By Lemma 3.2, and this completes the poof. P v 0 = d+/t d, 6
4 Lineaity : Pobability of having few backwad edges Now, we conside the moe geneal case when at most A backwad edges ae allowed. In this section, we get an exact expession fo the coesponding pobabiity. This estimate plays an impotant ole late in getting new and impoved lowe bounds on αh fo locally spase gaphs and linea hypegaphs. Ou goal in this section is to pove the following esult. Theoem 4. Fo a k-unifom linea hypegaph H, a vetex v having degee d, a unifomly chosen pemutation π induces at most A backwad edges with pobability P v A given by { if d A ; P v A = [ ta d ta+ A / d+/t ] if d A. Coollay 4.2 As d, the asymptotic expession fo the pobability P v A is given by /t A P v A = ΩA/d /t + /ta d Poof The asymptotics ae w..t. d, d A. The expession fo having at most A backwad edges is P v A = = dd...a + d A! + ta d A! d + /td + /t...a + + /t + ta + /td + td... + ta + Now, fo 0 < x, we have + x > e x. So we get P v A > + ta e /t d =A+ / = + ta e /t[ d = / A = /] = + ta /t[ln d ln A]+O/tdA e = + ta /t lnd/a O/tdA e = + ta A/d /t Ω = ΩA/d /t The above expession theefoe becomes ΩA/d /t. The vesion of PIE used most commonly deals with NT =, i.e. the numbe of elements in the set of inteest - in this case, pemutations of [td + ] which do not have any of the popeties unde consideation in this case, backwad edges with espect to v. Howeve we need something slightly diffeent - an expession fo the numbe of pemutations which have at least A backwad edges. Clealy, the emaining pemutations ae those which have at most A backwad edges. Theefoe, we use a slightly modified vesion of PIE, which is stated below in Theoem 4.5. This fom is well-known see e.g. [6], Chapte 2, Execise, although it seems to be used less fequently. Fo the sake of completeness, we povide a simple poof. Fist we state two identities involving binomial coefficients. The fist can be poved easily by induction on b and we omit the poof, and the second is poved in the appendix. 7
Lemma 4.3 Fo a, b nonnegative integes, b a + b a + i i a + i i i=0 = Lemma 4.4 Given non-negative integes d, A, d A and a positive intege t, d A + + A t + A + = We now pesent the genealized PIE and its well-known poof. At d A ta + d+/t Theoem 4.5 Let S be an n-set and E, E 2,...E d not necessaily distinct subsets of S. Fo any subset M of [d], define NM to be the numbe of elements of S in i M E i and fo 0 j d, define N j := M =j NM. Then the numbe N a of elements of S in at least a, 0 a d of the sets E i, i d, is Poof Take an element e S. d a a + i N a = i N i+a... MP IE i i=0 i Suppose e is in no intesection of at least a E i s. Then e does not contibute to any of the summands in the RHS of the expession MPIE, and hence, its net contibution to the RHS is zeo. ii Suppose e belongs to exactly a + j of the E i s, 0 j d a. Then its contibution to the RHS of MPIE is j a + j a + l l a + l l l=0 and by Lemma 4.3 this is equal to. Poof of Theoem 4. If d A, then P v A = obviously. The poof is simila to the poof of Theoem 3., except that in place of the PIE, we use Theoem 4.5. The set unde consideation is the set of pemutations of [td+], the subsets E i coespond to the pemutations fo which the i-th edge is backwad. It is easy to see that NM = NT M unde the notation used in Theoem 3. and hence NM = td+! t M +. Theefoe we have N j = d td+! j tj+ as befoe. Hence the expession fo the pobability Q v A that at least A edges ae backwad unde a unifomly andom pemutation π, becomes: Q v A = d A + i i i + A i ti + A +. i=0 By Lemma 4.4 the RHS of the above expession is Q v A = + ta 8 d A d+/t.
Hence the pobability of having at most A backwad edges is given by d P v A = A + ta d+/t and the poof is complete. 5 Lowe bounds fo linea hypegaphs and K -fee gaphs In this section we pove Theoems.2,.3, and.4. These follow by a simple application of Coollay 4.2. Since the poofs follow the same outline, we pove them simultaneously, highlighting only the diffeences as and when they occu. Poofs of Theoems.2,.3 and.4. Conside a unifomly chosen andom pemutation of the vetices of the gaph/hypegaph unde consideation. Let D be the aveage degee of the gaph o hypegaph and A = D ɛ. Let I be the set of those vetices each having at most A backwad edges incident on it. Clealy, the expected size of I is E[ I ] = P v A c /t A = ca /t max {A, dv} max {A, dv} v V v V fo some constant c = ck, ɛ. Fo a gaph, k = 2 and hence t =. Also, by constuction, the aveage degee of the subhypegaph induced by I is at most ka. Theefoe, thee exists an independent set I of size at least as follows i Case t =, gaph is K 3 -fee: By [2], αg is at least I Ω log2a = Ω log D 2A v V v V max {A, dv} ii Case t =, gaph is K -fee > 3: By [4], αg is at least log2a I log D Ω = Ω log log2a 2A log log D max {A, dv} iii Case t >, hypegaph is linea: By [8], αh is at least Ω log ka /t I = Ω log D /t ka /t v V The above thee cases pove Theoems.2,.3 and.4 espectively. v V max {A, dv} /t /t Note: An inspection of the poofs above show why we need ɛ to be a fixed constant. It is because all thee expessions above essentially have log A i.e. ɛ log D in the numeato. So, if ɛ = o, then log A = olog D, and we would get asymptotically weake esults. 9
6 Constuction compaing aveage degee vs. degee sequence based bounds A degee sequence-based bound obviously educes to a bound based on aveage degee, when the hypegaph is egula. Howeve, the convexity of the function x /t, x and t N, shows that the bounds in Theoems.2,.3 and.4 ae bette than the coesponding aveage degee-based bounds poved in [4], [2] and [4] espectively povided the minimum degee is at least A, although it is not clea a pioi if the impovement can become significantly lage. Also, at least half the vetices will have degee at most 2D, so even in the geneal case no estiction on the minimum degee ou bounds ae no wose than the aveage degee based bounds ignoing the constant factos. In fact, they can be much lage than the latte bounds. We now give infinite families of K -fee gaphs and linea k-unifom hypegaphs which show that i The bounds given by Theoem.2,.3 can be bette than the bounds in [2, 2, 4] espectively by a multiplicative facto of log V G. ii The bound in Theoem.4 can be bette than the bound in [4] by a multiplicative facto of log V H /log log V H ɛ/t, whee ɛ is the constant mentioned in Theoem.4. Case i Take a set of n disjoint gaphs, K,, K 2,2, K 4,4,..., K 2 n,2n. Fo each i [n], join one of the pats of the component K 2i,2 i to one of the pats in K 2 i,2i, by intoducing a complete bipatite gaph between them. Use the othe pat of K 2 i,2i fo joining to K 2 i+,2i+. Let G denote the esulting connected tiangle-fee gaph. The total numbe of vetices is 2 n+ 2, wheeas the aveage degee is d av = 2 EG / V G = 2 n + /2 o. Hence, the aveage degee based bound gives Θ V G log d av /d av = Θlog d av. Denote by l the maximum j such that 3.2 j A < 3.2 j+, whee A := d ɛ av. Fo evey fixed ɛ 0,, we have n l = Θn. Theoem.2 gives c log d av v V max{dv, A} = c log d av A + l j= 3.2 j A = clog d av [Θ + Θn] = clog d av Θlog V G n 2 + j=l+ 3.2 j 3.2 j + 2n 2 n The same example woks fo Theoem.3 also, since tiangle-fee gaphs ae obviously K -fee, fo 3. Case ii Fix some m = mn = k 2n. Fo each i {0,..., n }, fist ceate a connected linea hypegaph as follows: Take the vetex set as [k] 2i, i.e. the set of 2 i -dimensional vectos with each co-odinate of a vecto taking values in {, 2,..., k}. Let each hypeedge consists of the k vetices which have all but one co-odinate fixed. Call this hypegaph an i-unit. It can be veified easily that each i-unit is k-unifom and 2 i -egula. Now fo each i, ceate an i-component as follows: i Take m i = m k 2i disjoint unions of i-units and linealy ode them, say i,..., i mi. ii Conside the sets of vetices of size k fomed by choosing at most one vetex fom each i-unit. Add such edges geedily, ensuing the following: 0
i No vetex belongs to moe than one such edge; ii Choose the fist edge fom i-units i,..., i k, the second one fom i 2,..., i k+, etc. - in geneal the j-th such edge has one vetex fom each of the i-units i j mod mi, i j+ mod mi,... i j+k mod mi. Each i-component is a connected, linea k-unifom gaph and the degee of evey vetex is eithe 2 i o 2 i +. Take the disjoint unions of n such i-components, one fo evey i {0,..., n }, to get the hypegaph H = Hn = V, E. Fo each j {0,..., n 2}, geedily add a maximal matching between components j and j +, with each edge taking only one vetex fom component j and emaining k fom the component j +, and no vetex belonging to moe than one such edge. Let G be the esulting connected, linea k- unifom gaph. The total numbe of vetices in the j-th component is m j k 2j = m+o, and hence V = nm + o. Also, the aveage degee is d av 2 n /n 2 n /n. Let l denote the geatest intege j such that 2 j d av ɛ 2 ɛn /n ɛ. Theefoe the aveage degee based bounds in [4, 8] give a lowe bound of αh = Ωmn +/t log d av /t /2 n/t... A Notice that the degee of any vetex in the i-th component afte G has been constucted is always between 2 i and 2 i + 3. Fo a vetex v such that dv < d ɛ av, the actual degee does not play a ole in the expession in Theoem.4. Fo vetices v such that dv d ɛ av, this incease is negligible 2 ɛn /n ɛ + 3 2 ɛn /n ɛ. Theefoe, the bound in Theoem.4 gives l αh = Ω log d av /t = Ω j=0 mn ɛ/t 2 ɛn/t + n m 2 j/t j=l+ [ mlog d av /t ɛ2 ɛn/t n +ɛ/t + 2 ɛn/t n ɛ/t 2 n l /t 2 /t [ = Ω mlog d av /t 2 ɛn/t = Ω ɛn +ɛ/t + n ɛ/t ] 2 n l /t 2 /t... B mlog d av /t 2 ɛn/t ɛn +ɛ/t + Θn ɛ/t The atio of the bound in B to the one in A can be seen to be Ω2 n /n ɛ/t, which is Ωlog V / log log V ɛ/t. ] 7 Binomial Identities In the couse of this pape, cetain non-tivial binomial identities wee also obtained, with semicombinatoial poofs. Some of the identities ae new, to the best of ou knowledge, and may be of independent inteest. These ae descibed below: A d a a=0 i=0 d a + i a + i 2A + 2 i 2d 2a i!2a + i! = d! 2 4 A + i A 3
The LHS when divided by 2d +! amounts to the expession fo P v A when k = 3: choose a + i hypeedges fom the d hypeedges incident on v, of these a hypeedges ae backwad, while i hypeedges each have one vetex occuing pio to v in the andom pemutation. These i vetices can be chosen fom i pais in 2 i ways. The 2a + i vetices befoe v can be aanged in 2a + i! ways amongst themselves. The emaining 2d 2a i vetices occu afte v and can be aanged amongst themselves in 2d 2a i! ways. The RHS is easily obtained fom Theoem 4. by taking t = 2. Even the A = 0 case of the above identity gives afte some eaangements: d d + i 2 i = 2 d d i=0 The above identity meits discussion in some detail in [0] Chapte 5, eqs. 5.20, 5.35-8; a nice combinatoial poof of it is povided in [7]. The next identity fo the moe geneal case k 3 is much moe complicated: A d a a=0 i=0 t j= ij=i;ij 0 d a + i i a + i t t a, i,..., i t 2 i2 t... t it ta + i + 2i 2 +... t i t!td ta i... t i t! [ / ] d d + /t = td +! + ta + t A + d A 4 The LHS again follows by simila aguments as fo 3, this time fo geneal t. Thee ae a backwad edges, i edges which have one vetex befoe v, i 2 edges with 2 vetices befoe v, and so on. The RHS follows fom Theoem 4.. Ou poof techniques fo identities 3, 4 involving PIE, ae non-standad. It may be an inteesting poblem in Enumeative Combinatoics to come up with combinatoial poofs of the identities 3, 4. In paticula, fo 4, it would be inteesting to come up with poofs using any standad technique such as induction, geneating functions, the WZ method etc. 8 Concluding Remaks As the constuctions of Section 6 show, ou degee-sequence-based lowe bounds can be asymptotically bette than the pevious aveage-degee-based bounds. This is in spite of using the pevious bounds in the poof. The powe of the andom pemutation method lies in that it allows us to obtain a elatively lage spase induced subgaph, ove which the application of the aveage-degee bound yields a much bette esult than a staightfowad application ove the entie gaph would have. With egad to the tightness of ou esults and the weakening paamete A, fistly, fom the poof of Theoems.2-.4, it is clea that ɛ = log A/ log D has to be at least a constant. Ideally, we may want to have ɛ = 0 in the bounds of Theoems.2,.3 and.4. The following example, 2
howeve, shows that it is possible to constuct a tiangle-fee gaph fo which the bound in say, Theoem.2 would give a value moe than the numbe of vetices: Take a disjoint union of A = K n/3,n/3 and B = K n/3, and intoduce a pefect matching between B and one of the pats of A. Now, V = n, D 2n/9, and hence if ɛ = 0, Theoem.2 would give a lowe bound of Ωn log n, which is asymptotically lage than V. Simila examples can be constucted with linea hypegaphs also. Acknowledgement The authos would like to thank the anonymous efeees fo thei emaks and suggestions, which helped in impoving the quality of this pape. The fist and thid authos would also like to thank N.R. Aavind fo some initial helpful discussions. Refeences [] M. Ajtai, P. Edős, J. Komlós and E. Szemeédi: On Tuán s Theoem fo Spase Gaphs. Combinatoica 4 98, 33-37. [2] M. Ajtai, J. Komlós and E. Szemeédi: A Note on Ramsey Numbes. J. Comb. Theoy Se. A 29 980 354-360. [3] N. Alon, J. Kahn and P.D. Seymou: Lage induced degeneate subgaphs Gaphs Combinat. 3 987 203-2. [4] M. Ajtai, J. Komlós, J. Pintz, J.H. Spence and E. Szemeédi: Extemal uncowded hypegaphs, J. Combinatoial Theoy Se. A 32, 32-335 982. [5] N. Alon and J.H. Spence 2008. The Pobabilistic Method, Wiley Intenational. [6] Y. Cao: New esults on the independence numbe. Technical Repot, Tel Aviv Univesity, 979. [7] Yai Cao and Zsolt Tuza: Impoved Lowe Bounds on k- Independence, J. Gaph Theoy 99, Vol. 5, 99-07. [8] R. Duke, H. Lefmann and V. Rödl: On uncowded hypegaphs, Random Stuctues & Algoithms, 6, 209-22, 995. [9] R. Bopanna: [Comment on Lance Fotnow s blog, aticle by Bill Gasach] http://blog.computationalcomplexity.og/200/08/today-is-paul-tuans-00thbithday.html, comment # 200. [0] Ronald L. Gaham, Donald E. Knuth, and Oen Patashnik: Concete Mathematics Reading, Massachusetts: Addison-Wesley, 994 [] J. Komlós, J. Pintz and E. Szemeédi: A lowe bound fo Heilbonn s poblem, J. London Math. Soc. 2 25 982, no., 3-24. [2] J.B. Sheae: A note on the independence numbe of tiangle-fee gaphs. Discete Math. 46 983 83-87. [3] J.B. Sheae: A note on the independence numbe of tiangle-fee gaphs II. J. Combintoial Theoy Seies B 2 300-307. [4] J.B. Sheae: On the independence numbe of spase gaphs. Random Stuctues & Algoithms 7 995 269-27. 3
[5] J. Spence, Tuán s theoem fo k-gaphs, Discete Mathematics 2 972 83 86. [6] R.P. Stanley: Enumeative Combinatoics, 997, Cambidge Univesity Pess. [7] Tamás Lengyel: A combinatoial identity and the wold seies. SIAM Review Vol. 35, No. 2 Jun. 993, pp. 294-297. [8] Tosten Thiele: A lowe bound on the independence numbe of abitay hypegaphs. J. Gaph Theoy Vol. 30, 3, Mach 999, pp. 23-22. [9] P. Tuán: On an extemal poblem in gaph theoy [in Hungaian]. Math Fiz. Lapok 48 94 436-452. [20] V.K. Wei, A lowe bound on the stability numbe of a simple gaph. Technical Memoandum TM 8-27-9, Bell Laboatoies, 98. 9 Appendix Poof of Lemma 4.4 Let the LHS be denoted by S d. Then, using the identity n n n = +, we have S d = [ d + A ] d A + + + A d A + = S d + + A t + ta + t + ta + since d d = 0. Now the second sum can be simplified as T d = = = d A + + A d! d A!A! d A d d A A ta + t + ta + t + ta + t/ta + + By Lemma 3.2, we get T d = ta + [ d A / ] d A + ta + /t d A Theefoe, S d = S d + ta + [ d A / ] d + /t d A 4
Unaveling the ecusion and noticing that S A = /ta +, we get that S d = /ta + = /ta + [ d A [ A + A / d + /t d A ] / ] A + /t + by evesing the ode of summation. Finally, the following claim completes the poof. Claim. Fo d A, t 0, ta + A + A A+/t+ = ta ta + d A d+/t Poof of Claim. We use induction on d. When d = A, the LHS is ta +, while the RHS is At ta+, so we have equality. Now assume equality fo d and conside the LHS fo d + : + ta + = At ta + = = = A + A+/t+ [ d A At ta + d++/t + At ta + d++/t + At ta + d++/t + / ] [ / ] d + /t + At + d d + + /t d A d A + d A + [ ] d d + + /t A d A + d At d A + [ ] d + d + d + td + At A d A + A d + A which is the equied expession on the RHS. 5