Ewan Davies, Matthew Jenssen, Will Perkins, Barnaby Roberts On the average size of independent sets in triangle-free graphs

Transcription

1 Ewan Davies, Matthew Jenssen, Will Perkins, Barnaby Roberts On the average size of inepenent sets in triangle-free graphs Article (Accepte version) (Referee) Original citation: Davies, Ewan an Jenssen, Matthew an Perkins, Will an Roberts, Barnaby (207) On the average size of inepenent sets in triangle-free graphs. Proceeings of the American Mathematical Society, 46 (). pp ISSN DOI: 0.090/proc/ American Mathematical Society This version available at: Available in LSE Research Online: February 208 LSE has evelope LSE Research Online so that users may access research output of the School. Copyright an Moral Rights for the papers on this site are retaine by the iniviual authors an/or other copyright owners. Users may ownloa an/or print one copy of any article(s) in LSE Research Online to facilitate their private stuy or for non-commercial research. You may not engage in further istribution of the material or use it for any profit-making activities or any commercial gain. You may freely istribute the URL ( of the LSE Research Online website. This ocument is the author s final accepte version of the journal article. There may be ifferences between this version an the publishe version. You are avise to consult the publisher s version if you wish to cite from it.

2 ON THE AVERAGE SIZE OF INDEPENDENT SETS IN TRIANGLE-FREE GRAPHS EWAN DAVIES, MATTHEW JENSSEN, WILL PERKINS, AND BARNABY ROBERTS Abstract. We prove an asymptotically tight lower boun on the average size of inepenent sets in a triangle-free graph on n vertices with maximum egree, showing that an inepenent set rawn uniformly at ranom from such a graph has expecte size at least ( + o ()) log n. This gives an alternative proof of Shearer s upper boun on the Ramsey number R(3, k). We then prove that the total number of inepenent sets in a triangle-free [ ( graph with maximum egree is at least exp + o 2 () ) log 2 ] n. The constant /2 in the exponent is best possible. In both cases, tightness is exhibite by a ranom -regular graph. Both results come from consiering the har-core moel from statistical physics: a ranom inepenent set I rawn from a graph with probability proportional to λ I, for a fugacity parameter λ > 0. We prove a general lower boun on the occupancy fraction (normalize expecte size of the ranom inepenent set) of the har-core moel on triangle-free graphs of maximum egree. The boun is asymptotically tight in for all λ = O (). We conclue by stating several conjectures on the relationship between the average an maximum size of an inepenent set in a triangle-free graph an give some consequences of these conjectures in Ramsey theory.. Introuction.. Inepenent sets in triangle-free graphs. Ajtai, Komlós, an Szemeréi [] prove that any triangle-free graph G on n vertices with average egree has an inepenent set of size at least c log n for a small constant c. Shearer [23] later improve the constant to, asymptotically as, showing that such a graph has an inepenent set of size at least log + f() n where f() = = ( + o ( ) 2 ()) log. Here, an in what follows, logarithms will always be to base e. We use stanar asymptotic notation, with subscripts inicating which parameters the implie functions epen on. For example we write o () for a quantity that tens to zero as tens to infinity. The off-iagonal Ramsey number R(3, k) is the least integer n such that any graph on n vertices contains either a triangle or an inepenent set of size k. The above result of Ajtai, Komlós, an Szemeréi an a result of Kim [7] show that R(3, k) = Θ(k 2 / log k). Shearer s result gives the current best upper boun, showing that R(3, k) (+o())k 2 / log k. Inepenent work of Bohman an Keevash [6] an Fiz Pontiveros, Griffiths, an Morris [2] shows that R(3, k) (/4 + o())k 2 / log k. Reucing the factor 4 gap between these bouns is a major open problem in Ramsey theory. We prove a lower boun on the average size of an inepenent set in a triangle-free graph of maximum egree, matching the asymptotic form of Shearer s result, an in turn giving an alternative proof of the above upper boun on R(3, k). Date: March 6, 207.

3 2 EWAN DAVIES, MATTHEW JENSSEN, WILL PERKINS, AND BARNABY ROBERTS Theorem. Let G be a triangle-free graph on n vertices with maximum egree. Let I(G) be the set of all inepenent sets of G. Then I ( + o ()) log I(G) n. I I(G) Moreover, the constant is best possible. This result is weaker than Shearer s [23] in that instea of average egree we require maximum egree. Our result is stronger in that we show that the average size of an inepenent set from such a graph is of size at least (+o ()) log n, while Shearer shows the largest inepenent set is of at least this size (by analyzing a ranomize greey algorithm). The proof of Theorem is an aaptation of the authors metho use in [] to give a strengthening of results of Kahn [5] an Zhao [26] that a -regular graph has at most as many inepenent sets as a isjoint union of K, s on the same number of vertices. We make this explicit in Section 2, giving a unifie metho for proving these classical results of Shearer an Kahn. After sharing a raft of this paper with colleagues, we iscovere that James Shearer also knew the proof of the lower boun in Theorem an presente a sketch of it at the SIAM Conference on Discrete Mathematics in 998, but never publishe it [22]. To see that Theorem irectly implies the upper boun ( + o())k 2 / log k on R(3, k), suppose that G is triangle free with no inepenent set of size k. Then G must have maximum egree less than k. Applying Theorem we see the inepenence number is at least ( + o k ()) log k k n but less than k, an so n < ( + o k()) k2 log k as require. Of course this reasoning simply uses the average size of an inepenent set as a lower boun for the maximum size. In Section 5 we consier whether the iscrepancy between the maximum an average size can be exploite to improve the upper boun on R(3, k). After bouning the average size of an inepenent set from G, we next give lower bouns on the total number of inepenent sets in G, I(G). Theorem 2. Let G be a triangle-free graph on n vertices with maximum egree. Then I(G) e ( 2 +o ()) log2 n. Moreover, the constant /2 in the exponent is best possible. In comparison, improving on previous results of Cooper an Mubayi [9], Cooper, Dutta, an Mubayi [8] prove that any triangle-free graph of average egree has at least e ( 4 +o ()) log2 n inepenent sets. As a simple corollary we get the following lower boun without egree restrictions. Corollary. Let G be a triangle-free graph on n vertices. Then I(G) e ( ) 2 log 2 n +o() 4 log n. This improves on a result of Cooper, Dutta, an Mubayi [8] by a factor of 2 in the exponent; they also provie a construction base on the analysis of the triangle-free process in [6, 2] showing that the optimal constant is at most + log (as compare to the constant 2 log in Corollary ).

4 ON THE AVERAGE SIZE OF INDEPENDENT SETS 3 In Section 4 we show that the constants in Theorem an /2 in Theorem 2 are best possible, using the ranom -regular graph (with fixe as the number of vertices tens to infinity) as an example. It remains an interesting open question to etermine whether these constants are tight when the egree of the graph grows with the number of vertices; for example, if = n c for some c (0, )..2. The har-core moel. Theorems an 2 are special cases of results on a more general moel from statistical physics; the har-core istribution on the inepenent sets I of a graph G. For more on the har-core moel an its links with statistical physics an combinatorics, see [7]. The istribution epens on a fugacity parameter λ > 0 an is given by Pr[I] = λ I J I λ J. The enominator, P G (λ) = J I λ J, is the partition function of the har-core moel on G (also calle the inepenence polynomial of G). If λ =, the partition function P G () is the total number of inepenent sets an the har-core istribution is simply the uniform istribution over all inepenent sets of G. In what follows G will always be a graph on n vertices. We write α(g) for the size of the largest inepenent set in G. The expecte size of an inepenent set rawn from the har-core moel on G at fugacity λ is () α G (λ) := I I I Pr[I] = I λ I = λp G (λ) P G (λ) P G (λ) = λ (log P G(λ)). I I The key result of this paper is the following general lower boun on α G (λ) for triangle-free graphs. The lower boun is written naturally in terms of the Lambert W function, W (z): for z > 0, W (z) enotes the unique positive real satisfying the relation W (z)e W (z) = z. It will be useful to note that for z e we have W (z) log z log log z. Theorem 3. Let G be a triangle-free graph with maximum egree. Then for any λ > 0, (2) n α G(λ) λ + λ W ( log( + λ)). log( + λ) The quantity n α G(λ), the expecte fraction of vertices of G in the ranom inepenent set, is known as the occupancy fraction of G at fugacity λ. It is interesting to note that the lower boun in Theorem 3 is not monotone in λ whereas for any graph G, α G (λ) is monotone increasing (see Proposition ). Simply substituting λ = into (3), oes not quite suffice to prove Theorem. Surprisingly it turns out to be better to use a smaller λ an then appeal to the monotonicity of α G (λ). As an example, using λ = / log in (3) is enough to prove Theorem. This shows that Theorem hols even when we replace the average size of an inepenent set with a weighte average biase towar small sets. In fact we can affor to bias using any λ of the form o(). For smaller λ the lower boun in Theorem 3 becomes asymptotically weaker. For example taking λ = s where s (0, ) is fixe, (3) gives α G (λ) ( s + o ()) log n. This lower boun is tight an we in fact show that Theorem 3 (extene by monotonicity) is asymptotically tight for all λ = O () in Section 4. Turning to Theorem 2 an the problem of counting inepenent sets, we again give a more general statement about the har-core moel. We boun the partition function of a

5 4 EWAN DAVIES, MATTHEW JENSSEN, WILL PERKINS, AND BARNABY ROBERTS triangle-free graph at any fugacity λ using the fact that the occupancy fraction is the scale logarithmic erivative of the partition function. Inee from (.2) it follows that (3) n log P G(λ) = n λ 0 α G (t) t Using the lower boun on the occupancy fraction from Theorem 3 in the integral above gives the lower boun on the partition function which we state below, noting that Theorem 2 is a irect consequence. Theorem 4. Let G be a triangle-free graph on n vertices with maximum egree. Then for all λ > 0, ( [W P G (λ) exp ( log( + λ)) 2 + 2W ( log( + λ)) ] n ). 2 In particular, taking λ =, we see that G has at least e ( 2 +o ()) log2 n inepenent sets. t. 2. Lower bouns on the occupancy fraction The proof of Theorem 3 begins in the same way as the triangle-free case of the proof in [] that the occupancy fraction of the har-core moel on any -regular graph is at most that of K,. The main ifference is that in the last step we ask for the minimum instea of the maximum of the same constraine optimization problem. The metho is similar to that of Shearer [24] in lower bouning the inepenence number of K r -free graphs an that of Alon [3] in lower bouning the inepenence number of graphs of average egree in which neighborhoos are r-colorable. Alon an Spencer [4] use this technique to give a result like Theorem but with a worse constant. See also chapters 2 an 22 of [9] for an exposition of some of these results, an [8] for a recent application to hypergraphs. Unlike in previous works, in this paper we use the full power of choosing the fugacity parameter λ, rather than consiering only the uniform istribution. There are two key steps: First, we efine a ranom variable that epens on two sources of ranomness: the ranom inepenent set rawn from the har-core moel on G an a uniformly chosen ranom vertex v V (G). We then express the occupancy fraction in terms of two ifferent expectations involving this ranom variable. This gives a constraint on the istribution of the ranom variable. We then optimize over all ranom variables that satisfy the constraint, an euce a boun on the occupancy fraction. Recall that α G (λ) is the expecte size of an inepenent set rawn from the har-core moel on G at fugacity λ, an n α G(λ) is the occupancy fraction. In places where no confusion shoul arise we omit G from the notation. Proof of Theorem 3. Let I be an inepenent set of G rawn from the har-core moel at fugacity λ. For any vertex v in G, consier I 0 = I \ N(v) (noting that I 0 may inclue v itself). We say a vertex v is occupie if v I an unoccupie otherwise. Furthermore we say v is uncovere if N(v) I = an covere otherwise. Note that if v is covere, it must be unoccupie. We begin by stating some basic properties of the har-core moel. Fact : Pr[v I v uncovere] = λ +λ. This follows from the observation that for any realization of I 0 uner the event that v is uncovere, there are exactly two possibilities for I: I = I 0 an I = I 0 {v}, since v is uncovere an so may be ae.

6 ON THE AVERAGE SIZE OF INDEPENDENT SETS 5 Fact 2: Pr[v uncovere v has j uncovere neighbors] = ( + λ) j. First note that the event that v is uncovere is the same as the event that its uncovere neighbors are all unoccupie. The probability that an uncovere neighbor of v is unoccupie is +λ by Fact. Moreover, since G is triangle free the uncovere neighbors of v form an inepenent set an so they are in fact unoccupie inepenently with probability (4) (5) We now write the occupancy fraction as: n α(λ) = Pr[v I] n v G = λ + λ n Pr[v uncovere] v G = λ + λ n v G j=0 Pr[v has j uncovere neighbors] ( + λ) j where (2) follows from Fact an (2) from Fact 2. Now let Z be the ranom variable that counts the number of uncovere neighbors of a uniformly chosen vertex v. Z has two layers of ranomness, that of I rawn from the har-core measure, an that of selecting v at ranom. Interpreting the RHS of (2) in terms of Z, we obtain (6) n α(λ) = λ + λ E[( + λ) Z ]. An alternative way to relate the occupancy fraction to Z is to observe that in the sum v u v Pr[u I] each vertex u appears eg(u) times. Any uncovere neighbor of v is occupie with probability λ +λ by Fact an any covere neighbor is occupie with probability zero. Then since G has maximum egree we have (7) n α(λ) = Pr[v I] Pr[u I] = λ E Z n n + λ. v G We aim to minimize the occupancy fraction subject to the constraints on the istribution of Z given by (2) an (2). In fact we relax the optimization problem to optimize over all istributions of ranom variables Z that satisfy these constraints, not only those that arise from the har-core moel on a graph. In the calculation below we show that Jensen s inequality applie to (2) implies that the minimizer is achieve by the unique constant ranom variable Z that satisfies the constraints with equality. From (2) an (2) we have + λ λ α(λ) max n v u v { } E Z E Z, ( + λ) min x R + max { x, ( + λ) x}. To compute the minimum observe that the first of our lower bouns is increasing in x, an the secon is ecreasing. Then the minimum occurs at the value of x which makes the two bouns equal i.e. the x that satisfies an hence The result follows. xe log(+λ)x = log( + λ)x = W ( log( + λ)). +λ.

7 6 EWAN DAVIES, MATTHEW JENSSEN, WILL PERKINS, AND BARNABY ROBERTS A simple observation is the fact that lower bouns at a small fugacity λ imply the same bouns at higher fugacities. Proposition. For any graph G, the expecte size of an inepenent set α G (λ) is monotone increasing in λ. Proof. We will show that the erivative of α(λ) with respect to λ is positive. We write (using P for P G (λ)) ( ) λp α (λ) = = P P P + λp P λ(p ) 2 P ( 2 = P P + λ 2 P ( ) ) λp 2 λ P P = E( I ) + E( I 2 ) E( I ) (E( I )) 2 λ = Var( I ) 0 λ where I is a ranom inepenent set rawn from the har-core moel at fugacity λ. Now using Theorem 3 an Proposition we prove the main statement of Theorem. We leave the proof of tightness to Section 4. Proof of Theorem. Substituting λ = / log in (3) an recalling the boun W (z) log z log log z for z e we obtain n α G(λ) ( + o()) log. The result then follows from the monotonicity given by Proposition. So far we have been concerne with lower bouns on the occupancy fraction of a trianglefree graph of maximum egree. In [] the current authors consiere upper bouns, showing that the occupancy fraction of a -regular graph is at most that of K,. The proof metho is essentially ientical to the proof of Theorem 3. To make this connection explicit, we give a proof of the upper boun result here, restricte to the triangle-free case. Theorem 5 ([]). Let G be a -regular triangle-free graph, then for any λ > 0 n α G(λ) 2 α K, (λ) with equality only if G is a isjoint union of copies of K,. Proof. Note that since G is triangle free, equation (2) hols for G an since G is -regular, (2) hols with equality throughout. That is we have (8) n α G(λ) = λ E Z + λ = λ + λ E[( + λ) Z ], where we recall that Z is a ranom variable boune between 0 an. Now instea of asking for the minimum value of n α G(λ) over all istributions of Z as we i in the proof of Theorem 3, we ask instea for the maximum. Note that since 0 Z/ convexity of the function x ( + λ) x implies that (9) ( + λ) Z Z ( + λ) + Z.

8 ON THE AVERAGE SIZE OF INDEPENDENT SETS 7 Substituting this into (2) an using linearity of expectation yiels (0) n α G(λ) = λ E Z + λ λ( + λ) 2( + λ), where one can check that the right han sie is the occupancy fraction of K,. For uniqueness, note that to have equality in (2) we must have ha equality in (2) which is only possible if Z takes only the values 0 an. This istribution of Z can only occur in a isjoint union of copies of K,. To see this recall that Z is the number of uncovere neighbors of a ranomly selecte vertex v. The only way every vertex v can always have either 0 or uncovere neighbors is for all the neighbors of v to have the same neighborhoo. For -regular graphs this property hols only in graphs consisting isjoint unions of K,. Note that by the integral in (.2), Theorem 5 immeiately implies that for a triangle-free graph G, n log P G(λ) 2 log P K, (λ). Taking λ = implies that G has at most as many inepenent sets as a isjoint union of copies of K, on the same number of vertices. This last statement was famously prove by Kahn [5] (in the case where G is bipartite) using the entropy metho. 3. Counting inepenent sets In this section we prove the main statement of Theorem 4 (an hence Theorem 2) by integrating the lower boun on the occupancy fraction of a triangle-free graph given in Theorem 3. Again, the proof of tightness is eferre to Section 4. We also prove Corollary. Proof of Theorem 4. By (.2) an Theorem 3 we have () log P G (λ) n = n = n 2 λ W ( log( + t)) ( + t) log( + t) t 0 W ( log(+λ)) 0 ( + u) u [ W ( log( + λ)) 2 + 2W ( log( + λ)) ]. where for the first equality we use the substitution u = W ( log( + t)). In particular when λ =, using the inequality W (z) log z log log z for z e, we have ( ) log 2 log P G (λ) 2 + o () n. Proof of Corollary. In a triangle-free graph the neighborhoo of any vertex forms an inepenent set. Let be the largest egree of a vertex in G, then we have the boun { [ n P G (λ) max ( + λ), exp 2 W ( log( + λ))2]}, by consiering the neighborhoo of a vertex of maximum egree an by inequality (3). The first expression is increasing in while the secon is ecreasing, an at = ( ) n n log( + λ) 2 2 log( + λ) log 2

9 8 EWAN DAVIES, MATTHEW JENSSEN, WILL PERKINS, AND BARNABY ROBERTS they are equal. It follows that for λ > 0, [ ( ) ] n log( + λ) n log( + λ) (2) P G (λ) exp log Take λ = to complete the proof. Inequality (3) may be of inepenent interest, giving a general lower boun for the inepenence polynomial of a triangle-free graph on n vertices. 4. Ranom regular graphs Here we will show that the constants in Theorems to 4 are tight, as evience by the ranom -regular graph (conitione on having no triangles). To o this, we first leave the worl of finite graphs an consier the har-core moel on the infinite -regular tree T. The har-core moel can be efine on a infinite graph by taking the limit of har-core moels on a growing sequence of finite graphs (in this case -regular trees of increasing epth) with some bounary conitions specifie. There is a unique translation invariant har-core measure on the infinite -regular tree (see e.g. [6]), an uner this measure the probability that any given vertex is in the inepenent set is α T (λ), where α T (λ) is the solution of the equation (3) λ = α ( α) ( α 2α Using α T to enote the occupancy fraction of the infinite -regular tree is stanar, however we note that our notation for the analogous quantity n α G(λ) in a finite graph G graph on n vertices inclues the scaling /n. We compare the lower boun in Theorem 3 to α T (λ): Proposition 2. For every triangle-free graph G of maximum egree an any λ = O (), ). n α G(λ) ( + o ())α T (λ). In the language of statistical physics, α T (λ) is the replica symmetric occupancy fraction of a -regular graph of large girth. Extremality of a replica symmetric solution has been prove in several contexts incluing [0, 2, 25] but usually requires a conition such as an attractive potential function (e.g. the ferromagnetic Ising moel) or bipartiteness in the case of the har-core or monomer-imer moels. Proposition 2 states that in fact uner very minimal conitions (triangle free), the replica symmetric solution is a lower boun to the true occupancy fraction for all λ = O (), at least asymptotically in. This result can be compare with [, Theorem 2] where we use stronger assumptions, showing that a - regular, vertex-transitive bipartite graph has occupancy fraction strictly greater than α T (λ), though we only believe that result is best possible for λ below the uniqueness threshol on the tree, λ ( ) = Θ ( ( 2) ), in the sense that for any ɛ > 0, there exists some finite graph G with α G (λ) < α T (λ) + ɛ. Proof of Proposition 2. In what follows all asymptotic notation refers to the limit. We first show that α T (λ) = ( + o()) W (λ). We then show that for λ = o(), our boun on α G (λ) for triangle-free graphs given by Theorem 3 is asymptotically ( + o()) W (λ). We

10 ON THE AVERAGE SIZE OF INDEPENDENT SETS 9 then appeal to monotonicity (Proposition ) to exten this lower boun to all λ = O(). α T (λ) Letting z = 2α T (λ), equation (4) efining α T (λ) becomes ( (4) z + ) z = λ. First note that from (4) it follows easily that z = O(log ) an so ( + ) z = ( + o())e z. It then follows that z = W (λ + o(λ)). Using the fact that W x = W (x) x(+w (x)) W (x) x along with the Mean Value Theorem, we euce z = ( + o())w (λ). By the efinition of z we then have (5) α T (λ) = z ( + 2 z ) ( ) z z 2 = + O 2 Let us now suppose that λ = o(). By Theorem 3 we then have n α G(λ) = ( + o()) W (λ). λ W ( log( + λ)) = ( + o()) W (λ + O(λ2 )) + λ log( + λ) = ( + o()) W (λ), where for the last equality we use the same argument via the erivative of W as above. In particular if λ = / log say, we have that n α G(λ) ( + o()) log an by Proposition, the same hols when λ = Θ(). Now when λ = Θ(), α T (λ) = ( + o()) log by (4) an so the result follows. In the context of the proof technique of Section 2, we can unerstan the erivation of (4) by imposing two local constraints on the har-core moel: that every vertex has the same probability of being occupie, an that conitione on a vertex v not being occupie, the events that each of its neighbors are uncovere are inepenent events. Contrast this with our lower boun, which essentially says that the worst case for the occupancy fraction is if the number of these events that hol is always a constant; that is, the corresponing events are as negatively correlate as possible. Now if is large, then the sum of inepenent inicator ranom variables is highly concentrate so it is plausible that the most negatively correlate case is not far from the inepenent case. Proposition 2 makes this precise. Returning to the worl of finite graphs, we note that in fact α T (λ) is approximately achieve by finite graphs for a very large range of λ, an so Theorem 3 is asymptotically tight. Bhatnagar, Sly, an Tetali [5] give a precise escription of the local istribution of the har-core moel on a ranom -regular graph G,n for λ below a specifie value λ upper = +o() (just below the conensation threshol of the moel). They show that the local istribution converges to that of the unique translation invariant har-core measure on the infinite -regular tree escribe above, an so in particular for λ < λ upper, n α G,n (λ) = α T (λ) + o n (), whp as n (note that this is a much stronger notion of approximation than that of Proposition 2, with the error term tening to 0 with n instea of ). Then provie we conition on G,n being triangle free, which occurs with positive probability (epening only

11 0 EWAN DAVIES, MATTHEW JENSSEN, WILL PERKINS, AND BARNABY ROBERTS on ), a ranom -regular graph inee exhibits the tightness of Theorem 3. Let us quickly remark that when λ = o () one can get better asymptotic agreement than that given in Proposition 2. For example, when λ = s for some fixe 0 < s <, one can show that both the lower boun in Theorem 3 an α T (λ) can be written as W (λ) Theorem 3 is tight in a rather strong sense in this range. ( ) + O log an so +s An easy consequence of the results of [5] (either by their Sections 3.2 an 3.3 or by integrating the occupancy fraction) is that Theorem 4 is tight asymptotically in in the exponent. In particular, whp, P G,n () = e ( 2 +o ()) log2 n, showing that the constant in the exponent of Theorem 2 is tight. Finally let us note that for any fixe positive integer k, G,n in fact has girth at least k with positive probability (epening only on ). It follows that even if a stronger lower boun on girth is assume in Theorems to 4, the results cannot be improve asymptotically. 5. On the ratio of the maximum an average inepenent set size In light of the above result showing that the average size of an inepenent set in a trianglefree graph with maximum egree is at least ( + o ()) log n, we now raise the question of whether the largest inepenent set shoul be significantly larger. This gives a new way to pursue an upper boun on R(3, k). There is always a gap between the maximum an average size of an inepenent set (since the empty set is an inepenent set), but in general the ratio of maximum to average size can be arbitrarily close to. For example, the complete graph K n has maximum inepenent set size with average size n/(n + ). We conjecture that such a narrow gap cannot occur in triangle-free graphs. The following three conjectures make this claim precise in ifferent ways. Conjecture. For every triangle-free graph G, α(g) α G () 4/3. Replacing 4/3 with any number strictly greater than woul give an improvement to the R(3, k) boun. The graph with the smallest ratio α/α() we have foun is the triangle-free cyclic graph that exhibits the boun R(3, 9) 36 [4]. For this graph α/α() = = We choose 4/3 since it is a nice fraction less than.43 an since it is the ratio of maximum to average size in a triangle. One might woner if the extremal R(3, k) graphs are goo caniates for pushing the ratio α/α() own to. However, for large k it may be the case that graphs arising from the triangle-free process are asymptotically extremal, as is conjecture in [2]. We believe that for such graphs the ratio α/α() in fact converges to 2. This motivates the following conjectures. Conjecture 2. For every triangle-free graph G of minimum egree, α(g) α G () 2 o (). Conjecture 3. For every ɛ > 0, there exists λ > 0 so that for all triangle-free graphs G, α(g) α G (λ) 2 ɛ.

12 ON THE AVERAGE SIZE OF INDEPENDENT SETS Lemma. The following improvements to Shearer s upper boun on R(3, k) woul follow from the above conjectures an Theorems an 3. () Conjecture implies R(3, k) (3/4 + o())k 2 / log k. (2) Conjecture 2 implies R(3, k) (/2 + o())k 2 / log k. (3) Conjecture 3 implies R(3, k) (/2 + o())k 2 / log k. Proof. () an (3) are immeiate as Theorem 3 implies that for any fixe λ > 0, α G (λ) ( + o ()) log n in any maximum egree, triangle-free graph G on n vertices. To show (2) we take any triangle-free graph of maximum egree, greeily removing any vertices of egree at most an their neighbors. 2 log One possible approach to the above conjectures is via the following simple consequence of the proof of Proposition. For any graph G α(g) = lim λ α G(λ) = α G () + Var λ ( I ) λ In this paper we gave a lower boun for the expecte size of an inepenent set rawn from a triangle-free graph accoring to the har-core moel. The above equation shows that one approach to the above conjectures woul be o to the same for the variance. It is not clear that there is anything special about excluing triangles as oppose to other graphs, an so we make the following more general conjecture. Conjecture 4. For every K r -free graph G, α(g) α G () + r. For every K r -free graph G of minimum egree, with r fixe as. α(g) α G () 2 o (), To conclue, we give a lower boun on α/α(λ) that is tight for isjoint unions of r-cliques. The result also shows that α/α() is boune away from for all graphs containing an inepenent set of linear size. Theorem 6. For any graph G, or equivalently, (6) α(g) α G (λ) + α(g) λn, P (λ) ( λ α + ) P (λ). n This is tight when G is a union of r-cliques, for any r. Proof. Let P (λ) be the partition function of G. Define Q(λ) = P (λ) λ α P (λ) n P (λ). λ.

13 2 EWAN DAVIES, MATTHEW JENSSEN, WILL PERKINS, AND BARNABY ROBERTS Then Q(λ) P (λ) = λ P (λ) α P (λ) P (λ) n P (λ) = α(λ) α α(λ) λn, an so after rearranging, it suffices to show that Q(λ) 0 for all λ. In fact, we will show that all coefficients of Q(λ) as a polynomial in λ are non-negative. Let Q[k], enote the coefficient of λ k in Q(λ), an let i k = P [k] be the number of inepenent sets of size k in G. Then So it is enough to show for k =,..., α that Q[k ] = i k k α i k k n i k. (7) k i k i k n n(k ). α The following inequality is ue to Moon an Moser [20] (though it is usually state for cliques instea of inepenent sets): ( (8) k 2 k 2 i ) k(g) i k (G) n i k+(g) i k (G). We procee by inuction on k, from k = α own to k =. If k = α, then (5) gives: which is exactly (5) with α = k. i k (G) i k (G) n k 2 Now inuctively assume i k+ i k n( k α ) k+, an plug this into the RHS of (5). Rearranging, this gives ( i k n i k k α + ), kα which is (5). The integrate form of (6) is a simple boun on the partition function in terms of the inepenence number an number of vertices: for any graph G on n vertices with inepenence number α, ( P G (λ) + λn ) α. α This boun is tight for unions of cliques of the same size an is implicit in the work of Moon an Moser. It appears explicitly in [3], generalizing the λ = case from [2]. Acknowlegements We thank Rob Morris for his helpful comments on this paper, Noga Alon an Benny Suakov for recalling James Shearer s 998 SIAM talk, an James Shearer for sharing his transparencies from that talk with us.

14 ON THE AVERAGE SIZE OF INDEPENDENT SETS 3 References [] Miklós Ajtai, János Komlós, an Enre Szemeréi, A note on Ramsey numbers, Journal of Combinatorial Theory, Series A 29 (980), no. 3, [2] Vlaimir E Alekseev, On the number of maximal inepenent sets in graphs from hereitary classes, Combinatorial-algebraic methos in iscrete optimization, University of Nizhny Novgoro (99), 5 8. [3] Noga Alon, Inepenence numbers of locally sparse graphs an a Ramsey type problem, Ranom Structures an Algorithms 9 (996), no. 3, [4] Noga Alon an Joel H Spencer, The probabilistic metho, 3r e., John Wiley & Sons, 20. [5] Nayantara Bhatnagar, Allan Sly, an Prasa Tetali, Decay of correlations for the harcore moel on the -regular ranom graph, Electronic Journal of Probability 2 (206). [6] Tom Bohman an Peter Keevash, Dynamic concentration of the triangle-free process, arxiv preprint arxiv: (203). [7] Graham Brightwell an Peter Winkler, Har constraints an the Bethe lattice: aventures at the interface of combinatorics an statistical physics, Proc. Int l. Congress of Mathematicians, vol. III, 2002, pp [8] Jeff Cooper, Kunal Dutta, an Dhruv Mubayi, Counting inepenent sets in hypergraphs, Combinatorics, Probability an Computing 23 (204), no. 04, [9] Jeff Cooper an Dhruv Mubayi, Counting inepenent sets in triangle-free graphs, Proceeings of the American Mathematical Society 42 (204), no. 0, [0] Péter Csikvári, Lower matching conjecture, an a new proof of Schrijver s an Gurvits s theorems, arxiv preprint arxiv: (204). [] Ewan Davies, Matthew Jenssen, Will Perkins, an Barnaby Roberts, Inepenent sets, matchings, an occupancy fractions, arxiv preprint arxiv: (205). [2] Gonzalo Fiz Pontiveros, Simon Griffiths, an Robert Morris, The triangle-free process an R(3, k), arxiv preprint arxiv: (203). [3] Davi Galvin, An upper boun for the number of inepenent sets in regular graphs, Discrete Mathematics 309 (2009), no. 23, [4] Charles M Grinstea an Sam M Roberts, On the Ramsey numbers R(3, 8) an R(3, 9), Journal of Combinatorial Theory, Series B 33 (982), no., [5] Jeff Kahn, An entropy approach to the har-core moel on bipartite graphs, Combinatorics, Probability an Computing 0 (200), no. 03, [6] Frank P Kelly, Stochastic moels of computer communication systems, Journal of the Royal Statistical Society. Series B (Methoological) (985), [7] Jeong Han Kim, The Ramsey number R(3, t) has orer of magnitue t 2 / log t, Ranom Structures & Algorithms 7 (995), no. 3, [8] Alexanr Kostochka, Dhruv Mubayi, an Jacques Verstraëte, On inepenent sets in hypergraphs, Ranom Structures & Algorithms 44 (204), no. 2, [9] Michael Molloy an Bruce Ree, Graph colouring an the probabilistic metho, vol. 23, Springer Science & Business Meia, 203. [20] JW Moon an Leo Moser, On a problem of Turán, Magyar Tu. Aka. Mat. Kutató Int. Közl 7 (962), [2] Nicholas Ruozzi, The Bethe partition function of log-supermoular graphical moels, Avances in Neural Information Processing Systems, 202, pp [22] James B Shearer, personal communication. [23], A note on the inepenence number of triangle-free graphs, Discrete Mathematics 46 (983), no., [24], On the inepenence number of sparse graphs, Ranom Structures & Algorithms 7 (995), no. 3, [25] Alan S Willsky, Erik B Suerth, an Martin J Wainwright, Loop series an Bethe variational bouns in attractive graphical moels, Avances in neural information processing systems, 2008, pp [26] Yufei Zhao, The number of inepenent sets in a regular graph, Combinatorics, Probability an Computing 9 (200), no. 02,

15 4 EWAN DAVIES, MATTHEW JENSSEN, WILL PERKINS, AND BARNABY ROBERTS Lonon School of Economics aress: University of Birmingham aress: