The growth constant of odd cutsets in high dimensions
TTHE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS
OHAD NOY FELDHEIM † AND YINON SPINKA ‡ Abstract.
A cutset is a non-empty finite subset of Z d which is both connected and co-connected.A cutset is odd if its vertex boundary lies in the odd bipartition class of Z d . Peled [18] suggestedthat the number of odd cutsets which contain the origin and have n boundary edges may be oforder e Θ( n/d ) as d → ∞ , much smaller than the number of general cutsets, which was shown byLebowitz and Mazel [15] to be of order d Θ( n/d ) . In this paper, we verify this by showing that thenumber of such odd cutsets is (2 + o (1)) n/ d . Introduction and results
We consider the integer lattice Z d as a graph with nearest-neighbor adjacency, i.e., the edge set isthe set of { u, v } such that u and v differ by one in exactly one coordinate. The edge-boundary of asubset U of Z d is the set of edges having exactly one endpoint in U , and the internal vertex-boundary of U is the set of vertices in U which are adjacent to some vertex outside U .A cutset is a non-empty finite subset of Z d which is both connected and co-connected (i.e., bothit and its complement span connected subgraphs). The edge-boundaries of cutsets are exactly thefinite minimal edge-cuts of Z d , i.e., finite minimal sets of edges whose removal disconnects Z d . Avertex of Z d is called odd (even) if it is at odd (even) graph-distance from the origin, and a subsetof Z d is called odd (even) if its internal vertex-boundary consists solely of odd (even) vertices. Inthis work, we study OddCut n ( d ), the number of odd cutsets in Z d with edge-boundary size n whichcontain the origin. Random samples of such sets are depicted in Figure 1. Our main result is thefollowing. Theorem 1.1.
There exists a constant
C > such that for any integer d ≥ and any sufficientlylarge multiple n of d , we have n d (cid:0) − d (cid:1) ≤ OddCut n ( d ) ≤ n d (cid:0) C log3 / d √ d (cid:1) . We further prove the existence of a growth constant for the number of odd cutsets.
Theorem 1.2.
For any integer d ≥ , the limit µ ( d ) := lim n →∞ OddCut dn ( d ) /n exists. The existence of the above limit is proven via a super-multiplicitivity argument. It follows fromTheorem 1.1 that µ ( d ) satisfies the following bounds:1 + 2 − d ≤ log µ ( d ) ≤ C log / d √ d . We remark that the divisibility condition on n in the theorems is essential as the size of the edge-boundary of an odd set in Z d is always a multiple of 2 d (see Lemma 1.3 below). † Stanford University. Department of Mathematics. Stanford, CA 94305, U.S.A. ‡ Tel Aviv University. School of Mathematical Sciences. Tel Aviv, 69978, Israel.
E-mail addresses : [email protected], [email protected] . Date : September 6, 2016.Research of Y.S. supported by Israeli Science Foundation grant 861/15, the European Research Council startinggrant 678520 (LocalOrder), and the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. a r X i v : . [ m a t h . C O ] S e p THE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS (a) n = 3000 (b) n = 600 Figure 1.
Samples of random odd cutsets in Z with n boundary edges.The lower bound in Theorem 1.1 is obtained with relative ease, by estimating the number of oddcutsets which are obtained as local fluctuations of a single set, bearing resemblance to a ( d − n is necessary, since any oddcutset S in Z d which contains the origin has at least 2 d (2 d −
1) boundary edges (see Corollary 1.4below).The upper bound in Theorem 1.1, which is the main result of this paper, is obtained by a moreinvolved method. It is based on the intuition that the primary phenomenon which accounts forthe number of odd cutsets is the great variety of possible local structures near the boundary. Inother words, every odd cutset can be obtained as a perturbation of one of a relatively small numberof global shapes. Thus, the proof of the upper bound is based on a classification of odd cutsetsaccording to their approximate global structure, which we call an approximation. We first showthat the number of different approximations is small and then provide tight bounds on the numberof regular odd sets corresponding to each approximation and use it to bound the total number ofodd cutsets. This general method of approximations goes back to Sapozhenko [20]. Similar methodswere used also by Peled [18], by Galvin and Kahn [8] and by the authors [4]. The proof given hererelies on ideas from [20] and follows the approach of [8] with simplifications introduced in [4] (in amore complex setting). As it requires no additional effort, we prove the upper bound under weakerconnectivity assumptions than those used in the definition of a cutset (see Theorem 4.1).We remark that although Theorem 1.1 is stated (and has meaningful content) for all d ≥
2, thebounds become crude when d is small, in which case a similar upper bound could be obtained fromthe bound in [15] for general cutsets.1.1. Discussion.
In 1988, Lebowitz and Mazel [15] investigated general cutsets in Z d (which theyrefer to as primitive Peierls contours ). They showed that the number of cutsets with boundarysize n which contain the origin is at most d n/d when d ≥
2, and used this to show that thelow-temperature expansion for the d -dimensional Ising model, written in terms of Peierls contours,converges when the inverse-temperature is at least 64(log d ) /d . Ten years later, Balister and Bol-lob´as [1] improved this result by reducing the aforementioned bound on the number of cutsets to(8 d ) n/d . They also proved that the number of such sets is bounded from below by ( cd ) n/ d . HE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS 3 (a)
The even and odd bipar-tition classes of Z d . (b) An odd set. (c)
A regular odd set.
Figure 2.
The vertices of Z d are depicted as squares, with the even vertices inwhite and the odd vertices in gray. An odd set is a set whose internal boundaryconsists solely of odd vertices. A set is regular if both it and its complement haveno isolated vertices.Odd cutsets have been used in various probabilistic models to obtain phase transition and torpidmixing results. Some of these include works on the hard-core model by Galvin [7], Galvin–Kahn [8],Galvin–Tetali [11, 12] and Peled–Samotij [19], on homomorphism height functions by Galvin [5] andon 3-colorings by Galvin [6], Galvin–Randall [10], Galvin–Kahn–Randall–Sorkin [9] and Peled [18](who also treated discrete Lipschitz functions). Recently, using a generalization of odd cutsets,the authors showed that the 3-state antiferromagnetic Potts model in high dimensions undergoesa phase transition at positive temperature.Peled [18] raised the question of whether or not the number of odd cutsets is of smaller order ofmagnitude than the total number of cutsets. Namely, he asked whether this quantity is of order d Θ( n/d ) or only of order e Θ( n/d ) . Theorem 1.1 resolves this question by showing that it is indeedthe latter and pinpointing the constant in the exponent, i.e., (2 + o (1)) n/ d .It is worthwhile to mention that the method of approximations, which we use to obtain our upperbound, played a role in many of the aforementioned works. This method goes back to Sapozhenkowho studied enumeration problems on bipartite graphs and posets [20, 21, 22] motivated by previousresults of Korshunov on antichains [13] and of Korshunov–Sapozhenko on binary codes [14].In addition to cutsets, other types of connected subgraphs of Z d have also been investigated. Inthis context, we mention the recent work of Miranda–Slade [17] who obtained estimates for thegrowth constant of lattice trees and lattice animals in high dimensions.1.2. Open problems.
The bounds obtained in Theorem 1.1 on the number of odd cutsets matchin the first order term at the exponent. The next order term is determined by λ ( d ) := log µ ( d ) − − d ≤ λ ( d ) ≤ C log / d √ d and it is natural to ask what the correct asymptoticsof λ ( d ) is. Namely, is it exponential as in the lower bound? Is it polynomial as in the upper bound?In [18], Peled also raised the question of determining the scaling limit of odd cutsets. He suggestedthat in contrast to the case of ordinary cutsets (without the oddness condition), where it is plausiblethat the scaling limit is super Brownian motion, it may be the case that a random odd cutsettypically contains a macroscopic cube in its interior.1.3. Notation.
Let G = ( V, E ) be a graph. For vertices u, v ∈ V such that { u, v } ∈ E , we saythat u and v are adjacent and write u ∼ v . For a subset U ⊂ V , denote by N ( U ) the neighbors of U , i.e., vertices in V adjacent to some vertex in U , and define for t > N t ( U ) := { v ∈ V : | N ( v ) ∩ U | ≥ t } . In particular, N ( U ) = N ( U ). Denote the internal and external vertex-boundary of U by ∂ • U := U ∩ N ( U c ) and ∂ ◦ U := N ( U ) \ U , respectively. We also use the notation ∂ •◦ U := ∂ • U ∪ ∂ ◦ U , THE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS U + := U ∪ ∂ ◦ U and v + := { v } + . The set of edges between two disjoint sets U and W is denoted by ∂ ( U, W ) := {{ u, w } ∈ E : u ∈ U, w ∈ W } . In particular, the edge-boundary of U is ∂U := ∂ ( U, U c ).We also write ∂u := ∂ { u } . The graph-distance between u and v is denoted by dist( u, v ). For twonon-empty sets U, W ⊂ V , we denote by dist( U, W ) the minimum graph-distance between a vertexin U and a vertex in W . Policy regarding constants.
In the rest of the paper, we employ the following policy onconstants. We write
C, c, C (cid:48) , c (cid:48) for positive absolute constants, whose values may change from lineto line. Specifically, the values of
C, C (cid:48) may increase and the values of c, c (cid:48) may decrease from lineto line.1.4.
Odd sets and isoperimerty.
We use Even (Odd) to denote the set of even (odd) vertices of Z d . Thus, a set U ⊂ Z d is odd if and only if ∂ • U ⊂ Odd and it is even if and only if ∂ • U ⊂ Even.We say that U is regular if both it and its complement contain no isolated vertices. Thus, a cutsetis regular if and only if it is not a singleton. Observe that U is odd if and only if (Even ∩ U ) + ⊂ U and that U is regular odd if and only if U = (Even ∩ U ) + and U c = (Odd ∩ U c ) + . See Figure .For a set U ⊂ Z d and a unit vector s ∈ Z d , we define the boundary of U in direction s to be ∂ s U := { v ∈ U : v + s / ∈ U } . A nice property of odd sets is that the size of their boundary is thesame in every direction. Lemma 1.3.
Let U ⊂ Z d be finite and odd. Then, for any unit vector s ∈ Z d , we have | ∂ s U | = | Odd ∩ U | − | Even ∩ U | = | ∂U | d . Proof.
Denote U s := { u + s : u ∈ U } . The first equality follows from | Even ∩ U | = | Odd ∩ U s | = | Odd ∩ U s ∩ U | = | Odd ∩ U | − | Odd ∩ U \ U s | = | Odd ∩ U | − | U \ U s | = | Odd ∩ U | − | ∂ s U | . The second equality now follows from the first, since | ∂U | = (cid:80) s (cid:48) | ∂ s (cid:48) U | = 2 d · | ∂ s U | . (cid:3) Corollary 1.4.
Let U ⊂ Z d be finite and odd. If U contains an even vertex then | ∂U | ≥ d (2 d − .Proof. Let u ∈ U be even. Since U is odd, we have u + ⊂ U . Thus, | ∂ s U | ≥ d −
1, where s ∈ Z d isany unit vector, and the corollary follows from Lemma 1.3. (cid:3) We conclude with a well-known isoperimetric inequality.
Lemma 1.5 ([4]) . Let U ⊂ Z d be finite. Then | ∂U | ≥ d · | U | − /d . Organization.
The rest of the paper is organized as follows. In Section 2, we show thatOddCut n ( d ) is almost super-multiplicitive and use this to prove Theorem 1.2. In Section 3, weprove the lower bound stated in Theorem 1.1. In Section 4, we state two propositions; Propo-sition 4.2 which bounds the number of odd cutsets approximated by a given approximation andProposition 4.3 which shows that a relatively small number of approximations are sufficient toapproximate every odd cutset in question. We then deduce the upper bound stated in Theorem 1.1from these propositions. Section 5 and Section 6 are dedicated to the proofs of Proposition 4.2 andProposition 4.3, respectively.1.6. Acknowledgments.
We wish to thank Ron Peled for suggesting the problem to us and foruseful discussions. 2.
Almost super-multiplicitivity
The main step in showing the existence of the limit defining the growth constant µ ( d ) is estab-lishing the following “almost” super-multiplicitivity property of OddCut n ( d ). HE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS 5 uv { x + x = a + 1 } w (a) Modifying an odd cutsetto create a peak. { x + x = 2 r + 2 s } v v u (b) Modifying an odd cutset(with a peak) to adjust itsboundary size. { x + x = u + u + 1 } u (c) Merging two odd cutsets(one with a peak and onewith an inverted peak).
Figure 3.
Modifying odd cutsets.
Proposition 2.1.
Let d ≥ and let n, m, k ∈ d N with k ≥ d . Then OddCut n + m + k ( d ) ≥ OddCut n ( d ) · OddCut m ( d ) (cid:0) md (cid:1) dd − . Proof.
Fix d ≥ C n the collection of odd cutsets S in Z d having | ∂S | = n andcontaining the origin so that |C n | = OddCut n ( d ). Endow Z d with the partial order induced bythe sum of the first two coordinates. We say that a vertex u is the peak of an odd set S if it isthe unique maximal element among all the even vertices in S . Denote by P n the collection of oddcutsets in C n having a peak and by O n those having a peak at the origin. The proof consists offour parts:(1) |C n | ≤ |P n +2 d (2 d − | .(2) |P n | ≤ |O n | · ( n/ d ) d/ ( d − .(3) |O n | ≤ |O n + k − d | .(4) |P n | · |O m | ≤ |C n + m − d | .Since we may assume that m ≥ d (2 d −
1) by Corollary 1.4, the proposition then follows from |C n | · |C m | (1) ≤ |P n +2 d (2 d − | · |P m +2 d (2 d − | (2) ≤ |P n +2 d (2 d − | · |O m +2 d (2 d − | · ( m/ d + 2 d − d/ ( d − ≤ |P n +2 d (2 d − | · |O m + k − d +10 d | · ( m/d ) d/ ( d − ≤ |C n + m + k | · ( m/d ) d/ ( d − . To prove the first part, we take a set S ∈ C n and construct a set S (cid:48) ∈ P n +2 d (2 d − in an injectivemanner. Let a be the maximum value for which the hyperplane { x + x = a } intersects S . Let v be a vertex in this intersection, let u ∈ S be a adjacent to v (note that S (cid:54) = { v } since + ⊂ S ) anddenote w := u + e + e / ∈ S . Since S is odd and v + e / ∈ S , we have that v is odd, u is even, u + ⊂ S and v − u ∈ { e , e } . It is easy to check that S (cid:48) := S ∪ w + is an odd set with | ∂S (cid:48) | = n + 2 d (2 d − w . Since S (cid:48) is clearly connected, it remains only to check that S (cid:48) is co-connected.Since S is co-connected, any two vertices x, y ∈ ( S (cid:48) ) c can be connected via a path in S c . Thus, itsuffices to check that any two vertices x, y ∈ ( S (cid:48) ) c ∩ N ( S (cid:48) \ S ) can be connected via a path in ( S (cid:48) ) c .Using that ( S (cid:48) ) c ∩ N ( S (cid:48) \ S ) = w ++ ∩ { x + x ≥ a + 1 } and that w is the peak of S (cid:48) , this is easilyverified. See Figure .The second part follows from the isoperimetric inequality in Lemma 1.5. THE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS
To prove the third part, let r, s ∈ N , and, for S ∈ O n , define S (cid:48) := S ∪ (cid:8) ( i, i, , . . . , ∈ Z d : 0 < i ≤ r (cid:9) + ∪ (cid:8) ( r + 2 i, r, , . . . , ∈ Z d : 0 < i ≤ s (cid:9) + . It is straightforward to check that S (cid:48) ∈ C n + r (2 d − s (2 d − and has a peak at ( r + 2 s, r, , . . . , S (cid:55)→ S (cid:48) is injective, this shows that |O n | ≤ |O n + r (2 d − s (2 d − | . See Figure .Finally, since 2 d − d − k − d > (2 d − d − r, s ∈ N so that k − d = r (2 d −
3) + s (2 d −
2) (this is known as Sylvester’ssolution to the Frobenius problem for two coins).To prove the fourth part, we take an element (
S, S (cid:48) ) ∈ P n × O m and construct a set T ∈ C n + m − d in an injective manner. Let S (cid:48)(cid:48) be the reflection of S (cid:48) through the hyperplane { x + x = 0 } (i.e., S (cid:48)(cid:48) is obtained by negating the first two coordinates of every vertex in S (cid:48) ). Let u be the peak of S and define T := S ∪ R , where R := u + e + e + S (cid:48)(cid:48) . Since S and R lie on different sides of thehyperplane { x + x = u + u + 1 } (except for { u + e , u + e } , which is their common intersectionwith this hyperplane), it follows easily that T is a connected odd set with | ∂T | = n + m − d . To seethat T ∈ C n + m − d , it remains to check that T is co-connected, i.e., that T c = S c ∩ R c is connected.This follows from the observation that S := S c ∩ { x + x ≤ u + u + 1 } is connected, and similarly,that R := R c ∩ { x + x ≥ u + u + 1 } is connected, and from S ∩ R (cid:54) = ∅ and T c = S ∪ R . Finally,it is clear that this mapping is injective, since u can be recovered by considering all hyperplanes { x + x = a } which intersect T at two points and using that | ∂S | is known. See Figure . (cid:3) Proof of Theorem 1.2.
Fix d ≥ a n := OddCut dn ( d ) and b n := a n − d / n . Clearly,it suffices to show the existence of the limit lim n →∞ b /nn . This will follow from Fekete’s subadditivelemma (applied to − log b n ) if we show that b n is a super-multipliciative sequence. Indeed, byProposition 2.1, for n ≥ m > d , we have b n + m = a ( n − d )+( m − d )+6 d n + m ) ≥ a n − d · a m − d n + m ) · (2 m ) d/ ( d − ≥ b n b m · n ( n + m ) ≥ b n b m . (cid:3) The lower bound
The proof of the lower bound in Theorem 1.1 is based on a simple counting argument. The ideaappeared already in [18] (see also [19]). Since the details have not appeared in print, we give ashort proof here. Let d ≥ m be a large even integer. We first prove the lower boundfor n := 2 d ( m d − + ( d − m d − ) directly by constructing a large family of odd cutsets having n boundary edges. We then use Proposition 2.1 to extend the lower bound to other values of n .For brevity, we shall employ the notation [ a, b ) := { a, . . . , b − } for integers a < b . The proof isaccompanied by Figure 4.Let B := Even ∩ [0 , m ) d − × { } and observe that B +0 is an odd cutset in Z d which contains theorigin. We now show that its edge-boundary size is n . Let ↑ = e d be the d -th standard basis vectorand recall that the boundary of U ⊂ Z d in direction ↑ is ∂ ↑ U = { v ∈ U : v + e d / ∈ U } , and that, byLemma 1.3, | ∂U | = 2 d | ∂ ↑ U | if U is odd. Let π : Z d → Z d − be the projection onto the first d − π ( U ) = π ( ∂ ↑ U ) for any finite set U . Observe also that π ( B +0 ) = [0 , m ) d − ∪ d − (cid:91) i =0 Odd d − ∩ [0 , m ) i × {− , m } × [0 , m ) d − − i , where Odd d − denotes the set of odd vertices in Z d − . Thus, since π is injective on ∂ ↑ B +0 , we have | ∂B +0 | d = | ∂ ↑ B +0 | = | π ( ∂ ↑ B +0 ) | = | π ( B +0 ) | = m d − + ( d − m d − = n d . Let A := Even ∩ [1 , m − d − × {± } and observe that for any B ⊂ A , the set ( B ∪ B ) + is an odd cutset. Since π (( B ∪ B ) + ) = π ( B +0 ) and since π is injective on ∂ ↑ ( B ∪ B ) + , we HE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS 7 * * * * * * * * * ×× ×× ×× ×× ×× ×× ×× ×× + ++ +
Figure 4.
Constructing a large family of odd cutsets. The vertices in B , A and A ( B ) (all of which are even vertices) are depicted by ∗ , × and +, respectively.The set B +0 is an odd cutset (shown in red), and any choice of B ⊂ A gives anodd cutset ( B ∪ B ) + with the same number of boundary edges (the red and pinkregions). Once B is chosen, any choice of B ⊂ A ( B ) then gives another oddcutset ( B ∪ B ∪ B ) + also with the same number of boundary edges (the red, pinkand gray regions).deduce that | ∂ ( B ∪ B ) + | = | ∂B +0 | = n . Also notice that ( v , . . . , v d − , ∈ ( B ∪ B ) + ifand only if ( v , . . . , v d − , ∈ B , and similarly, ( v , . . . , v d − , − ∈ ( B ∪ B ) + if and only if( v , . . . , v d − , − ∈ B , so that different choices of B produce distinct sets. By counting thenumber of such sets, we obtainOddCut n ( d ) ≥ | A | = 2 ( m − d − = 2 n d · ( m − d − md − d − md − ≥ n d · (1 − d/m ) . To obtain a better bound, we consider a “second order” augmentation. Given B ⊂ A , define A ( B ) := (cid:110) x ∈ Even ∩ [2 , m − d − × {± } : ∀ ≤ i ≤ d − x , . . . , x d − , x d ) ± e i ∈ B (cid:111) . As before, one may easily check that for every choice of B ⊂ A and B ⊂ A ( B ), the set( B ∪ B ∪ B ) + is a distinct odd cutset with precisely n boundary edges. In order to use this toimprove the lower bound, we apply a first moment argument. Let X be a uniformly chosen randomsubset of A and denote Y := | A ( X ) | . Then using Jensen’s inequality, we obtainOddCut n ( d ) ≥ (cid:88) B ⊂ A | A ( B ) | = 2 | A | · E (cid:2) Y (cid:3) ≥ | A | + E [ Y ] . By linearity of expectation, we have E [ Y ] = ( m − d − · − (2 d − so thatOddCut n ( d ) ≥ ( m − d − +( m − d − · − d +2 ≥ n d · (1 − d/m )(1+2 − d +2 ) ≥ n d · (1+2 − d +1 ) . (1)To complete the proof of the lower bound, we now extend this bound to arbitrary (large) n (cid:48) ∈ d N .Let m be the largest even integer satisfying n (cid:48) − d ≥ n := 2 d ( m d − + ( d − m d − ). Then, byProposition 2.1, by (1), by the maximality of m and since OddCut d (2 d − ( d ) = 1, we obtainOddCut n (cid:48) ( d ) ≥ d OddCut n ( d ) ≥ ( m − d − +( m − d − · − d +2 − log (4 d ) ≥ n (cid:48) d (1+2 − d +1 ) . The upper bound
For the upper bound in Theorem 1.1, which is the main result of this paper, we consider a slightlymore general class of sets. Recall that a subset of Z d is called regular if both it and its complementcontain no isolated vertices. For a graph G and a positive integer r , we denote by G ⊗ r the graphon the same vertex set as G in which two vertices are adjacent if their distance in G is at most r .A finite regular subset of Z d is an r -cutset if both it and its complement are connected in ( Z d ) ⊗ r .Note that the notions of a regular cutset and a 1-cutset coincide. We write S dn,r for the collectionof odd r -cutsets S in Z d having | ∂S | = n and dist( , S ) ≤ r , where denotes the origin in Z d . THE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS * * ** * * * ** * * * ** * ** * * * ** * * * ** ** * * ** * * ** * * * * * * ** * * * * ** ** * * * ** * * * ** (a)
An approximation A . * * ** * * * ** * * * ** * ** * * * ** * * * ** ** * * ** * * ** * * * * * * ** * * * * ** ** * * * ** * * * ** * * ** * * * ** * * * ** * ** * * * ** * * * ** ** * * ** * * ** * * * * * * ** * * * * ** ** * * * ** * * * ** (b) Two possible regular odd sets approximated by A . Figure 5.
An approximation and two regular odd sets approximated by it are illus-trated. Vertices belonging to A • ( A ◦ ) are known to be in S ( S c ); these are depictedin ( a ) by a red (yellow) background. The remaining vertices belong to A ∗ = ( A • ∪ A ◦ ) c and are unknown to be in S or S c ; these are depicted by ∗ and a white background. Theorem 4.1.
There exists a constant
C > such that for any integers d ≥ and n, r ≥ , |S dn,r | ≤ n d (cid:0) Cr log3 / d √ d (cid:1) . As discussed in the introduction, the proof is based on a classification of odd r -cutsets accordingto their approximate global structure. To this end, we require some definitions. An approximation is a pair A = ( A • , A ◦ ) of disjoint subsets of Z d such that A • is odd and A ◦ is even. We say that A approximates an odd set S if A • ⊂ S and A ◦ ⊂ S c . Thus, we think of A • as the set of verticesknown to be in S , A ◦ as the vertices known to be outside S , and A ∗ := ( A • ∪ A ◦ ) c as the verticeswhose association is unknown.Let 1 ≤ t < d be an integer. A t -approximation is an approximation A such that the subgraphof Z d induced by A ∗ has maximum degree at most t and has no isolated vertices. For an illustrationof these notions, see Figure 5. It is instructive to notice that if a t -approximation A approximates S , then any unknown vertex is near the boundary in the sense that A ∗ ⊂ ( ∂ •◦ S ) + ; see (2) below.We now give two key propositions which summarize the role of t -approximations in our proofof the upper bound. We henceforth fix the dimension d ≥ d in the notation. Denote by S the collection of regular odd sets and by S n the collection of S ∈ S having | ∂S | = n . For an approximation A , denote by S ( A ) the collection of S ∈ S whichare approximated by A . We extend this notation to a family of approximations A , by setting S ( A ) := ∪ A ∈A S ( A ). Our first proposition justifies our notions of approximation by bounding thenumber of regular odd sets approximated by a given t -approximation. Proposition 4.2.
For any integers n ≥ and ≤ t < d and any t -approximation A , we have | S n ( A ) | ≤ n/ (2 d − t ) . Our second proposition shows that a small family of t -approximations suffices to approximateevery set in S n,r . Proposition 4.3.
There exists a constant
C > such that for any integers n, r ≥ and ≤ t < d ,there exists a family A of t -approximations of size |A| ≤ exp (cid:16) Cnr (cid:0) log dd (cid:1) / + Cn log ddt (cid:17) such that every S ∈ S n,r is approximated by some element in A , i.e., S n,r ⊂ S ( A ) . HE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS 9
The proofs of Proposition 4.2 and Proposition 4.3 are given in Section 5 and Section 6, respec-tively. Equipped with these propositions, we are now ready to prove the upper bound.
Proof of upper bound in Theorem 4.1.
Let d ≥
2, 1 ≤ t < d and n, r ≥ A be afamily of t -approximations obtained by applying Proposition 4.3. By Proposition 4.2, |S n,r | ≤ (cid:88) A ∈A | S n ( A ) | ≤ |A| · n/ (2 d − t ) ≤ n d (cid:16) t d − t + Cr log / d √ d + C log dt (cid:17) . Substituting any integer t satisfying (cid:112) d/ log d ≤ t ≤ √ d log / d yields the theorem. (cid:3) Counting regular odd sets with a given approximation
In this section, we prove Proposition 4.2. That is, our goal is to prove an upper bound onthe number of regular odd sets with given boundary size which are approximated by a particular t -approximation. The proof is based on an analysis of minimal vertex-covers.We henceforth fix an integer 1 ≤ t < d and a t -approximation A = ( A • , A ◦ ). Recall our notation A ∗ := ( A • ∪ A ◦ ) c . For S ∈ S ( A ), define D • ( S ) := A ∗ ∩ ∂ • S = Odd ∩ A ∗ ∩ S,D ◦ ( S ) := A ∗ ∩ ∂ ◦ S = Even ∩ A ∗ ∩ S c , (2)where the first equality follows from Odd ∩ A ∗ ∩ S ⊂ N ( A ◦ ) ⊂ N ( S c ), which in turn uses the factsthat A • is odd and the maximum degree of A ∗ is strictly less than 2 d ; the second equality followssimilarly. Define also D ( S ) := D • ( S ) ∪ D ◦ ( S ) = A ∗ ∩ ∂ •◦ S. Two key properties of this definition are that S is determined by D ( S ) and that D ( S ) is a minimalvertex-cover of A ∗ (see Figure ). This is stated precisely in the following lemma.A vertex-cover of a graph G is a subset of vertices U ⊂ V ( G ) satisfying that every edge of G hasan endpoint in U . A vertex-cover is minimal if it is minimal with respect to inclusion. Denote byMC( G ) the set of all minimal vertex-covers of G . For a set V ⊂ Z d , we also write MC( V ) for theset of minimal covers of the subgraph of Z d induced by V . Lemma 5.1.
The map S (cid:55)→ D ( S ) is an injective map from S ( A ) to MC( A ∗ ) Proof.
Let S ∈ S ( A ) and denote D • := D • ( S ), D ◦ := D ◦ ( S ) and D := D • ∪ D ◦ . To see that themap is injective, it suffices to reconstruct S from D . In fact, we can reconstruct S both from D • and from D ◦ , separately. Indeed, as A • ⊂ S and A ◦ ⊂ S c , it follows thatOdd ∩ S = Odd ∩ A • ∪ D • and Even ∩ S c = Even ∩ A ◦ ∪ D ◦ , and since S is regular odd, S is determined by Odd ∩ S via S = (Odd ∩ S ) ∪ N d (Odd ∩ S ) and byEven ∩ S c via S = (Even ∩ S ) + = (Even \ (Even ∩ S c )) + .Next, we show that D is a vertex-cover of A ∗ . To this end, let u, v ∈ A ∗ be a pair of adjacentvertices, and assume without loss of generality that u is odd and v is even. Assume towardsobtaining a contradiction that neither u nor v belong to D , and observe that in this case u / ∈ S and v ∈ S , which is impossible since S is odd. Hence either u ∈ D or v ∈ D .Finally, we show that D is a minimal vertex-cover. To this end, let v ∈ D and assume towardsa contradiction that N ( v ) ∩ A ∗ ⊂ D . Assume without loss of generality that v is odd, so that v ∈ D • and N ( v ) ∩ A ∗ ⊂ D ◦ . Since A • is odd, v is odd and v / ∈ A • , we have N ( v ) ∩ A • = ∅ . Thus, N ( v ) ⊂ A ◦ ∪ D ◦ ⊂ S c , which is impossible since v ∈ S and S is regular. (cid:3) We require the following lemma from [4]. ** ** **** ** ** ** (a)
A region of unknown verticesin A ∗ . These vertices are denotedby ∗ (with odd vertices having agray background). The vertices in A • and A ◦ are shown in red andyellow, respectively. * * ** * * * * ** * * (b) An example of ( D • , D ◦ ) and its corresponding regular odd set.The property that D • ∪ D ◦ is a minimal vertex-cover of A ∗ is man-ifested in the figure by the fact that there are no two adjacent ∗ .The corresponding regular odd set is obtained by adding each ver-tex in D • to S and each vertex in D ◦ to S c , and then determiningthe remaining vertices according to their neighbors. Figure 6.
The figure illustrates the process of recovering S from D • and D ◦ . Lemma 5.2 ([4, Lemma 4.9]) . Let G be a finite graph and let { p v } v ∈ V ( G ) be non-negative numberssatisfying p u + p v ≤ for all { u, v } ∈ E ( G ) . Then (cid:88) U ∈ MC( G ) (cid:89) u ∈ U p u ≤ . Applying this with p v = 1 / v ∈ V ( G ), yields |U | ≤ max U ∈U | U | , for any U ⊂
MC( G ) . Hence, Lemma 5.1 implies that for any n ≥ | S n ( A ) | ≤ |{ D ( S ) : S ∈ S n ( A ) }| ≤ max S ∈S n ( A ) | D ( S ) | . Proposition 4.2 is now an immediate consequence of the following lemma.
Lemma 5.3.
For any S ∈ S ( A ) , we have | D ( S ) | ≤ | ∂S | d − t . Proof.
Let S ∈ S ( A ) and denote D • := D • ( S ), D ◦ := D ◦ ( S ) and D := D • ∪ D ◦ . Since A • is odd, A ◦ is even and A ∗ induces a subgraph of maximum degree at most t , we haveOdd ∩ A ∗ ⊂ N d − t ( A ◦ ) and Even ∩ A ∗ ⊂ N d − t ( A • ) . Thus, | D • | ≤ | ∂ ( D • ,A ◦ ) | d − t and | D ◦ | ≤ | ∂ ( D ◦ ,A • ) | d − t . Since ∂ ( D • , A ◦ ) and ∂ ( D ◦ , A • ) are disjoint subsets of ∂S (since A • ⊂ S and A ◦ ⊂ S c ), we have | D | = | D • ∪ D ◦ | ≤ | ∂S | d − t . (cid:3) Constructing approximations
This section is dedicated to the proof of Proposition 4.3. That is, we show that there exists a smallfamily A of t -approximations which covers S n,r in the sense that S n,r ⊂ S ( A ). The constructionof A is done in two steps, which we outline here. For an approximation A , recall the notation A ∗ := ( A • ∪ A ◦ ) c and say that | A ∗ | is the size of A . The first step is to construct a small family ofsmall approximations which covers S n,r . HE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS 11
Lemma 6.1.
For any integers n ≥ and r ≥ , there exists a family A of approximations, eachof size at most Cn (cid:112) (log d ) /d , such that S n,r ⊂ S ( A ) and |A| ≤ exp (cid:16) Cnr log / dd / (cid:17) . The second step is to upgrade an approximation to a small family of t -approximations whichcovers at least the same collection of regular odd sets. Lemma 6.2.
For any integers n, m ≥ and ≤ t < d and any approximation A of size m , thereexists a family A of t -approximations such that S n ( A ) ⊂ S ( A ) and |A| ≤ exp (cid:16) C log dd · (cid:0) m + nt (cid:1)(cid:17) . Lemma 6.1 and Lemma 6.2 are proved in Sections 6.2, and 6.3 below.
Proof of Proposition 4.3.
Applying Lemma 6.1, we obtain a family B of approximations, each of sizeat most m := Cn (cid:112) (log d ) /d , such that S n,r ⊂ S ( B ) and |B| ≤ exp( Cnrd − / log / d ). ApplyingLemma 6.2 to each approximation in B , we obtain a collection of families of t -approximations. Tak-ing the union over this collection, we obtain a family A of t -approximations such that S n,r ⊂ S ( A )and |A| ≤ |B| · exp( Cnd − log d · ( (cid:112) (log d ) /d + 1 /t )). The required bound follows. (cid:3) Preliminaries.
In this section, we gather some elementary combinatorial facts about graphswhich we require for the construction of approximations. For the purpose of these preliminaries,we fix an arbitrary graph G = ( V, E ) of maximum degree ∆.
Lemma 6.3.
Let U ⊂ V be finite and let t > . Then | N t ( U ) | ≤ ∆ t · | U | . Proof.
This follows from a simple double counting argument. t | N t ( U ) | ≤ (cid:88) v ∈ N t ( U ) | N ( v ) ∩ U | = (cid:88) u ∈ U (cid:88) v ∈ N t ( U ) N ( u ) ( v ) = (cid:88) u ∈ U | N ( u ) ∩ N t ( U ) | ≤ ∆ | U | . (cid:3) The next lemma follows from a classical result of Lov´asz [16, Corollary 2] about fractional vertexcovers, applied to a weight function assigning a weight of t to each vertex of S . Lemma 6.4.
Let S ⊂ V be finite and let t ≥ . Then there exists a set T ⊂ S of size | T | ≤ t | S | such that N t ( S ) ⊂ N ( T ) . The following standard lemma gives a bound on the number of connected subsets of a graph.
Lemma 6.5 ([2, Chapter 45]) . The number of connected subsets of V of size k + 1 which containthe origin is at most ( e (∆ − k . Recall that G ⊗ r is the graph on V in which two vertices are adjacent if their distance in G is atmost r . The next simple lemma was first introduced by Sapozhenko [20]. Lemma 6.6 ([20, Lemma 2.1]) . Let
S, T ⊂ V and let a, b be positive integers. Assume that S isconnected in G ⊗ a , dist( s, T ) ≤ b for all s ∈ S and dist( S, t ) ≤ b for all t ∈ T . Then T is connectedin G ⊗ ( a +2 b ) . The following lemma, based on ideas of Tim´ar [23], establishes the connectivity of the boundaryof subsets of Z d which are both connected and co-connected. Lemma 6.7 ([3, Proposition 3.1]) . Let U ⊂ Z d be connected and co-connected. Then ∂ •◦ U isconnected. Corollary 6.8.
Let r ≥ be an integer and let U ⊂ Z d be such that U and U c are connected in ( Z d ) ⊗ r . Then ∂ •◦ U is connected in ( Z d ) ⊗ r .Proof. Since U is connected in ( Z d ) ⊗ r , it suffices to show that ∂ •◦ B is connected in ( Z d ) ⊗ r forevery connected component B of U . Let C be the collection of connected components of B c . Since U c is connected in ( Z d ) ⊗ r , it suffices to show that ∂ •◦ W ∪ ∂ •◦ W (cid:48) is connected in ( Z d ) ⊗ r whenever W, W (cid:48) ∈ C satisfy dist(
W, W (cid:48) ) ≤ r . This follows from Lemma 6.7. (cid:3) Constructing small approximations.
This section is devoted to the proof of Lemma 6.1.That is, we construct a small family of approximations, each of size at most Cn (cid:112) (log d ) /d , suchthat S n,r ⊂ S ( A ). This is done in two steps. First, we show that for every regular odd set S , thereexists a small set U such that N ( U ) separates S , where we say that a set W separates S if everyedge in ∂S has an endpoint in W . Lemma 6.9.
Let n ≥ be an integer and let S ∈ S n . Then there exists U ⊂ ( ∂ •◦ S ) + of size atmost Cnd − / √ log d such that N ( U ) separates S . We then show that every separating set gives rise to a small family of small approximations.
Lemma 6.10.
For any integer n ≥ and any finite W ⊂ Z d , there exists a family A of approx-imations, each of size at most | W | , such that every S ∈ S n which is separated by W satisfies S ∈ S ( A ) , and |A| ≤ | W | /d . Before proving these lemmas, let us show how they imply Lemma 6.1.
Proof of Lemma 6.1.
Let n, r ≥ S n is non-empty then n ≥ d .Thus, we may assume that n ≥ d . Denote k := Cnd − / √ log d , V := { + ie : 0 ≤ i < n } and V := { v ∈ Z d : dist( v, V ) ≤ r + 2 } . Let U be the collection of all subsets of Z d of size at most k which intersect V and are connected in ( Z d ) ⊗ ( r +4) . Since the maximum degree of ( Z d ) ⊗ ( r +4) is atmost ( Cd ) r +4 , Lemma 6.5 implies that |U | ≤ | V | · ( e ( Cd ) r +4 ) k ≤ exp (cid:16) Cnrd − / log / d (cid:17) , where the rightmost inequality uses the fact that | V | ≤ n (2 d + 1) r +2 and n ≥ d .For each U ∈ U , apply Lemma 6.10 to W = N ( U ) to obtain a family A U of approximations,each of size at most 3 | N ( U ) | ≤ dk , such that every S ∈ S n which is separated by N ( U ) satisfies S ∈ S ( A U ), and |A U | ≤ k . Denote A := ∪ U ∈U A U and note that A is a family of approximations,each of size at most 6 dk , such that |A| ≤ |U | · k ≤ exp (cid:16) Cnrd − / log / d (cid:17) . It remains to check that S n,r ⊂ S ( A ). Towards showing this, let S ∈ S n,r . By Lemma 6.9, thereexists U ⊂ ( ∂ •◦ S ) + of size at most k such that N ( U ) separates S . Thus, since S ∈ S ( A U ) bydefinition, to conclude that S ∈ S ( A ), it suffices to show that U ∈ U . Since | U | ≤ k , we need onlyshow that U is connected in ( Z d ) ⊗ ( r +4) and that U intersects V .We first show that U intersects V , or equivalently, that dist( U, V ) ≤ r + 2. Since N ( U ) separates S , we have ∂ •◦ S ⊂ N ( U ) + so that it suffices to show that dist( ∂ •◦ S, V ) ≤ r . Indeed, if / ∈ S then dist( , ∂ •◦ S ) ≤ r , since dist( , S ) ≤ r , and if ∈ S then V ∩ ∂ •◦ S (cid:54) = ∅ , since, by Lemma 1.5, | S | ≤ | ∂S | ≤ n .We are left with showing that U is connected in ( Z d ) ⊗ ( r +4) . Indeed, since U ⊂ ( ∂ •◦ S ) + , wesee that dist( u, ∂ •◦ S ) ≤ u ∈ U , and since ∂ •◦ S ⊂ N ( U ) + , we have dist( w, U ) ≤ w ∈ ∂ •◦ S . As S and S c are connected in ( Z d ) ⊗ r , Corollary 6.8 and Lemma 6.6 imply that U isconnected in ( Z d ) ⊗ ( r +4) . (cid:3) HE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS 13 vuz w (a) u, w, z ∈ S , v ∈ S c vuz w (b) v ∈ A , ( v, w ) ∈ G vuz w (c) w ∈ A , ( v, w ) ∈ G Figure 7.
Constructing the separating set. In (a) , a revealed vertex u ∈ S isdepicted along with a neighbor z ∈ S . Every four-cycle ( u, v, w, z ) such that v ∈ S c (and hence w ∈ S ) falls into one of two types. Either v has at least s boundaryedges as shown in (b) , or w has at least 2 d − s boundary edges as in (c) . At least1 / u isadjacent to many vertices which have many boundary edges and one such neighborof u is included in B ; if it the second type, then z is adjacent to many vertices whichhave almost all their edges in the boundary and z is included in B (cid:48) . The set U S isobtained by taking the union of B and B (cid:48) . Constructing separating sets.
Before proving Lemma 6.9, we start with a basic geometricproperty of odd sets which we require for the construction of the separating set.
Lemma 6.11.
Let S be an odd set and let { u, v } ∈ ∂S . Then, for any unit vector e ∈ Z d , either { u, u + e } or { v, v + e } belongs to ∂S . In particular, | ∂u ∩ ∂S | + | ∂v ∩ ∂S | ≥ d. Proof.
Assume without loss of generality that u is odd. Since S is odd, we have u ∈ S and v / ∈ S .Similarly, if u + e ∈ S then v + e ∈ S . Thus, either { u, u + e } ∈ ∂S or { v, v + e } ∈ ∂S . (cid:3) For a set S , denote the revealed vertices in S by S rev := { v ∈ Z d : | ∂v ∩ ∂S | ≥ d } . That is, a vertex is revealed if it sees the boundary in at least half of the 2 d directions. Thefollowing is an immediate corollary of Lemma 6.11. Corollary 6.12.
Let S be an odd set. Then S rev separates S .Proof of Lemma 6.9. Let n ≥ S ∈ S n . Note that ∂S = ∂S c implies that S rev = ( S c ) rev .Thus, in light of Corollary 6.12, it suffices to show that, for each R ∈ { S, S c } , there exists a set U R ⊂ N ( ∂ • R ) such that R ∩ R rev ⊂ N ( U R ) and | U R | ≤ Cnd − / √ log d . Indeed, the lemma thenfollows by taking U := U S ∪ U S c . Since S and S c are symmetric (up to parity), we may considerthe case R = S . The proof is accompanied by Figure 7.Denote s := √ d log d and t := d/
4, and define A := ∂ ◦ S ∩ N s ( ∂ • S ) and A (cid:48) := ∂ • S ∩ N d − s ( ∂ ◦ S ) . Observe that, by Lemma 6.3, | A | ≤ ns and | A (cid:48) | ≤ n d − s . We now use Lemma 6.4 with A to obtain a set B ⊂ A ⊂ ∂ ◦ S such that | B | ≤ dt | A | and N t ( A ) ⊂ N ( B ) . We also define B (cid:48) := S ∩ N t ( A (cid:48) ). By Lemma 6.3, we have | B (cid:48) | ≤ st | A (cid:48) | . Finally, we define U S := B ∪ B (cid:48) . Clearly, U S ⊂ N ( ∂ • S ) and | U S | ≤ n log dts + 2 snt (2 d − s ) ≤ n log dd √ d log d + 8 n √ d log dd = 20 n √ log dd / . It remains to show that S ∩ S rev ⊂ N ( U S ). Towards showing this, let u ∈ S ∩ S rev = ∂ • S ∩ N d ( ∂ ◦ S ).Since S is regular, there exists a vertex z ∈ N ( u ) ∩ S . Let F denote the set of pairs ( v, w ) suchthat ( u, v, w, z ) is a four-cycle and v ∈ ∂ ◦ S , and note that | F | ≥ d −
1. Denote G := { ( v, w ) ∈ F : v ∈ A } and G (cid:48) := { ( v, w ) ∈ F : w ∈ A (cid:48) } . Note that, by Lemma 6.11, F = G ∪ G (cid:48) and, for any ( v, w ) ∈ F , we have w ∈ S . Since F = G ∪ G (cid:48) ,either | G | or | G (cid:48) | is at least | F | / ≥ t . Now observe that if | G | ≥ t then u ∈ N t ( A ) ⊂ N ( B ), whileif | G (cid:48) | ≥ t then z ∈ N t ( A (cid:48) ) so that u ∈ N ( B (cid:48) ). Therefore, we have shown that u ∈ N ( U S ). (cid:3) From separating sets to small approximations.
Proof of Lemma 6.10.
Let n ≥ W ⊂ Z d . Consider the set X := Z d \ W . Say that aconnected component of X is small if its size is at most d , and that it is large otherwise.Let S ∈ S n be such that W separates S and observe that every connected component of X is entirely contained in either S or S c . Thus, if we let B • and B ◦ be the union of all the largecomponents of X which are contained in S and S c , respectively, we have that B • ⊂ S and B ◦ ⊂ S c .To obtain an approximation of S from ( B • , B ◦ ), define A • := B • ∪ (Odd ∩ B + • ) and A ◦ := B ◦ ∪ (Even ∩ B + ◦ ). Clearly, A • is odd and A ◦ is even, and, since S is odd, A • ⊂ S and A ◦ ⊂ S c . Hence, A = A ( S ) := ( A • , A ◦ ) is an approximation of S , i.e., S ∈ S ( A ).Next, we bound the size of A . For this we require a simple corollary of Lemma 1.5. Namely, | ∂T | ≥ d · min { d, | T |} for any finite T ⊂ Z d . Indeed, this follows immediately from Lemma 1.5 since 2 ≥ e /e ≥ x /x for all x >
0. Denotingby Y the union of all the small components of X , we observe that A ∗ ⊂ Y ∪ W . Since any smallcomponent T of X has | T | ≤ | ∂T | /d and ∂T ⊂ ∂W , we obtain | Y | ≤ | ∂W | d ≤ d | W | d ≤ | W | . Thus, | A ∗ | ≤ | Y ∪ W | ≤ | W | .Now, denote by A the collection of approximations A ( S ) constructed above for all S ∈ S n whichare separated by W . To conclude the proof, it remains to bound |A| . Let (cid:96) be the number of largecomponents of X , and observe that |A| ≤ (cid:96) . Since any large component T of X has | ∂T | ≥ d and ∂T ⊂ ∂W , we obtain (cid:96) ≤ | ∂W | /d ≤ | W | /d so that |A| ≤ | W | /d , as required. (cid:3) Constructing t -approximations. In this section, we take a small approximation and refineit into a multitude of t -approximations. The following lemma allows us to eliminate any isolatedunknown vertices in an approximation. Lemma 6.13.
For every approximation A there exists an approximation B such that S ( A ) = S ( B ) , B ∗ ⊂ A ∗ and B ∗ has no isolated vertices.Proof. The lemma clearly holds when S ( A ) = ∅ so that we may assume that S ( A ) (cid:54) = ∅ . Define B • := A • ∪ N d ( A • ) and B ◦ := A ◦ ∪ N d ( A ◦ ). Using the definition of a regular set and the assumptionthat S ( A ) is non-empty, it is straightforward to check that B = ( B • , B ◦ ) is an approximation andthat S ( B ) = S ( A ). Finally, since A • is odd and A ◦ is even, we have that the set of isolatedvertices of A ∗ is A ∗ \ N ( A ∗ ) = N d ( A ◦ ) ∪ N d ( A • ), and similarly for B ∗ . Thus, B ∗ has no isolatedvertices. (cid:3) HE GROWTH CONSTANT OF ODD CUTSETS IN HIGH DIMENSIONS 15
For an approximation A and an integer m ≥
0, we define S ∗ m ( A ) := (cid:8) S ∈ S ( A ) : | Odd ∩ A ∗ ∩ S | + | Even ∩ A ∗ ∩ S c | ≤ m (cid:9) . Note that, by Lemma 5.3 and (2), if A is a t -approximation for some 1 ≤ t < d then S n ( A ) ⊂ S ∗(cid:98) n/ (2 d − t ) (cid:99) ( A ) . (3) Lemma 6.14.
For any integers m ≥ and ≤ t < d and any approximation A , there exists afamily A of t -approximations such that S ∗ m ( A ) ⊂ S ( A ) and |A| ≤ exp( C log d · m/t ) .Proof. We may assume S ∗ m ( A ) (cid:54) = ∅ as otherwise the statement is trivial. Moreover, by Lemma 6.13we may assume that A ∗ has no isolated vertices. For an indepedent set W ⊂ A ∗ (i.e., a setcontaining no two adjacent vertices), write W Even := W ∩ Even and W Odd := W ∩ Odd, and define W • := W +Even ∪ (cid:0) Odd ∩ N t ( A ∗ \ W + ) (cid:1) ,W ◦ := W +Odd ∪ (cid:0) Even ∩ N t ( A ∗ \ W + ) (cid:1) . Here one should think of W as recording the locations of a subset of even vertices in A ∗ ∩ S and oddvertices in A ∗ ∩ S c . We shall see that if this subset is chosen suitably then W • ⊂ S and W ◦ ⊂ S c .Let W denote the family of such sets W having size at most m/t , and define B := { ( A • ∪ W • , A ◦ ∪ W ◦ ) : W ∈ W} . It is straightforward to check that every B ∈ B is an approximation. Let us show that, for any B ∈ B , the maximal degree of the subgraph induced by B ∗ is less than t , i.e., that B ∗ ∩ N t ( B ∗ ) = ∅ .Indeed, letting W ∈ W be such that B = ( A • ∪ W • , A ◦ ∪ W ◦ ) and noting that B ∗ = A ∗ \ ( W • ∪ W ◦ )and W • ∪ W ◦ = W + ∪ N t ( A ∗ \ W + ), we have B ∗ ∩ N t ( B ∗ ) ⊂ ( A ∗ \ N t ( A ∗ \ W + ) (cid:1) ∩ N t ( A ∗ \ W + ) = ∅ , Hence, applying Lemma 6.13 to every element in B , we obtain a family A of t -approximations suchthat |A| ≤ |B| and S ( A ) = S ( B ).Next, we bound the size of A . By assumption, there exists a set S ∈ S ∗ m ( A ). Since S is odd andsince A ∗ has no isolated vertices, we have A ∗ ⊂ (Odd ∩ A ∗ ∩ S ) + ∪ (Even ∩ A ∗ ∩ S c ) + , so that | A ∗ | ≤ (2 d + 1) m ≤ dm , and hence, |A| ≤ |B| ≤ |W| ≤ (cid:98) m/t (cid:99) (cid:88) k =0 (cid:18) dmk (cid:19) ≤ ( m/t + 1)(3 edt ) m/t ≤ d Cm/t . It remains to show that S ∗ m ( A ) ⊂ S ( B ). Let S ∈ S ∗ m ( A ) and let W be a maximal subset of A ∗ among those satisfying W Even ⊂ S , W Odd ⊂ S c and | A ∗ ∩ N ( W ) | ≥ t | W | . Observe that A ∗ ∩ N ( W ) ⊂ (Odd ∩ A ∗ ∩ S ) ∪ (Even ∩ A ∗ ∩ S c ) . Thus, t | W | ≤ | A ∗ ∩ N ( W ) | ≤ m and, as W is clearly an independent set, we have W ∈ W . Nowdefine B • := A • ∪ W • and B ◦ := A ◦ ∪ W ◦ so that B := ( B • , B ◦ ) ∈ B . We are left with showing that S ∈ S ( B ), i.e., that B • ⊂ S and B ◦ ⊂ S c . The two statements are very similar so we only show theformer. Since S ∈ S ( A ), it suffices to show that A ∗ ∩ W • ⊂ S . Let v ∈ A ∗ ∩ W • . If v ∈ W +Even then v ∈ S by the definition of W and since S is odd. Otherwise, v ∈ Odd ∩ ( A ∗ \ W + ) ∩ N t ( A ∗ \ W + )so that | A ∗ ∩ N ( W ∪ { v } ) | ≥ | A ∗ ∩ N ( W ) | + t . Hence, v ∈ S by the maximality of W . (cid:3) We are now ready to prove Lemma 6.2.
Proof of Lemma 6.2.
Applying Lemma 6.14 with m := m and t = d to A , we obtain a fam-ily B of d -approximation such that |B| ≤ exp( C log d · m/d ) and S ∗ m ( A ) ⊂ S ( B ). ApplyingLemma 6.14 with m := (cid:98) n/d (cid:99) and t := t to each B ∈ B , we obtain a collection of families of t -approximations. Taking the union over this collection, we obtain a family A of t -approximationssuch that S ∗(cid:98) n/d (cid:99) ( B ) ⊂ S ( A ) and |A| ≤ |B| · exp( C log d · n/dt ) ≤ exp( C log d · ( m/d + n/dt )) . It remains to show that S n ( A ) ⊂ S ( A ). To this end, let S ∈ S n ( A ) and note that S ∈ S ∗ m ( A ) ⊂S ( B ). Thus, there exists B ∈ B such that S ∈ S ( B ). Hence, S ∈ S n ( B ) and (3) now implies that S ∈ S ∗(cid:98) n/d (cid:99) ( B ) ⊂ S ( A ), as required. (cid:3) References
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