On the structure of subsets of the discrete cube with small edge boundary

The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F} \subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G} \subset \{0,1\}^n$ such that $|\mathcal{F} \Delta \mathcal{G}| \leq Cl$, where $C$ is an absolute constant. This is best-possible, up to the value of $C$. Our result can be seen as a `stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli concerning the isoperimetric inequality in Euclidean space.


Introduction
Isoperimetric inequalities are of ancient interest in mathematics. In general, an isoperimetric inequality gives a lower bound on the 'boundary size' of a set of a given 'size', where the exact meaning of these words varies according to the problem. One of the best-known examples is the isoperimetric inequality for Euclidean space, which states (informally) that among all subsets of R n of given volume (whose surface area is defined), a Euclidean ball has the smallest surface area. An exact formulation (actually, one of the versions) of this inequality is as follows.
areas such as communication complexity (see e.g. [17]), network science (see [3]) and game theory (see [18]). Hereafter, if A ⊂ {0, 1} n , we write ∂ A for the edge boundary of A with respect to Q n .
The edge isoperimetric problem for Q n was solved by Harper [17], Lindsey [27], Bernstein [2], and Hart [18]. Let us describe the solution. We may identify {0, 1} n with the power-set P ([n]) of  Let us describe the extremal families for Theorem 1.3. If F, G ⊂ P([n]), we say that F and G are weakly isomorphic if there exists an automorphism φ of Q n such that G = φ (F); in this case, we write F ∼ = G. Equivalently, F, G ⊂ {0, 1} n are weakly isomorphic iff G can be obtained from F by permuting the coordinates 1, 2, . . . , n and interchanging 0's with 1's on some subset of the coordinates. Clearly, weak isomorphism preserves the size of the edge boundary. It is well-known (and easy to check by analyzing known proofs of Theorem 1.3) that equality holds in Theorem 1.3 if and only if F is weakly isomorphic to L. In particular, if |F| is a power of 2, then equality holds in Theorem 1.3 if and only if F is a subcube.
As observed in [6], this result is best possible (except for the condition 0 ≤ δ < c, which was conjectured to be unnecessary in [6]). However, the problem of obtaining a sharp stability result for sets not of size a power of 2, remained open.

Our result
In this paper, we obtain the following stability result for the edge isoperimetric inequality in the discrete cube, which applies to families of arbitrary size and which is sharp up to an absolute constant factor. Theorem 1.5. There exists an absolute constant C > 0 such that the following holds. If F ⊂ P ([n]) and L ⊂ P([n]) is the initial segment of the lexicographic ordering with |L| = |F|, then there exists a family G ⊂ P([n]) weakly isomorphic to L, such that This is sharp up to the value of the absolute constant C. In fact, we conjecture that Theorem 1.5 holds with C = 2, due to the following example. Example 1.6. Let s,t, n be integers with t ≥ 2 and t + 2 ≤ s ≤ n, and let It is easy to see that for each of the above families F. We proceed to verify this when s = n and t = 2, i.e. for the family It is easy to check that |∂ (F n,n,2 )| − |∂ L| = 2. Now let G ⊂ P ([n]) be any family weakly isomorphic to L. Since L is contained in a codimension-1 subcube of P ([n]), so is G. However, it is clear that |F n,n,2 \ C| ≥ 2 for any codimension-1 subcube C of P ([n]). Hence, using the fact that |F n,n,2 | = |G|, we have |F n,n,2 ∆G| ≥ 2|F n,n,2 \ G| ≥ 2 min where the minimum is over all codimension-1 subcubes C of P ([n]). Clearly, we have |F n,n,2 ∆L| = 4. The last two facts imply (1.1).
We note that the relation between |∂ F| − |∂ L| and |F∆G| in Theorem 1.4 (which applies in the special case where |F| is a power of 2) is sharper than in Theorem 1.5, but the above example demonstrates that Theorem 1.5 is sharp (up to an absolute constant factor) in its more general setting.
Instead of the Fourier-analytic techniques used in most previous works on isoperimetric stability, our techniques are purely combinatorial. As is often the case with theorems concerning Q n , we prove Theorem 1.5 by induction on n, but the techniques we use in the inductive step are somewhat novel. The inductive step relies on an 'intermediate' structure theorem (Proposition 4.1) concerning the intersections of F with codimension-1 and codimension-2 subcubes, where F ⊂ P([n]) is a family with small edge boundary. This proposition is proved using some intricate combinatorial arguments, including shifting operators (a.k.a. 'compressions'), and a detailed analysis of the influences of the family (see below).

Related work
The edge boundary ∂ F of a subset F ⊂ P ([n]) is closely connected with the influences of F. For F ⊂ P ([n]), the ith influence of F is defined by . Note that I[F] = |∂ F|/2 n−1 -the total influence of a set is none other than the size of its edge boundary, appropriately normalised.
It is natural to rephrase this definition in the language of Boolean functions. If f : {0, 1} n → {0, 1}, we define the ith influence of f to be the probability that, if x ∈ {0, 1} n is chosen uniformly at random and the ith entry of x is flipped, the value of the function f changes. There is of course a natural one-to-one correspondence between subsets of P ([n]) and Boolean functions on {0, 1} n : for each F ⊂ P ([n]), we associate to F the Boolean function Over the last thirty years, many results have been obtained on the influences of Boolean functions (and functions on more general product spaces), and have proved extremely useful in such diverse fields as theoretical computer science, social choice theory and statistical physics, as well as in combinatorics (see, e.g., the survey [22]). One of the most useful such results (and one of the first major results on influences) is the seminal 'KKL theorem' (Kahn,Kalai and Linial [21]), which states that for any Boolean where c 0 is an absolute constant -so a Boolean function of expectation 1/2 has some coordinate of 'fairly large' influence, viz., Ω((log n)/n Here, to be completely formal, if G ⊂ P ([n]) we say that G depends upon k coordinates if there exists S ⊂ [n] with |S| = k, such that (T ∈ G) ⇔ (T ∩ S ∈ G) holds for all T ⊂ [n]. Friedgut's theorem implies that any F ⊂ P ([n]) with bounded total influence (at most K, say), and with measure |F|/2 n bounded away from 0 and 1, can be closely approximated by a 'junta' -that is, by a family which depends upon a bounded number of coordinates (depending on K). Friedgut's proof in [11] uses Fourier analysis and hypercontractivity, in a similar way to the proof of the KKL theorem. (We remark that in [8], Falik and Samorodnitsky gave different, more combinatorial proofs of the KKL theorem and of Friedgut's theorem, utilising martingales rather than Fourier analysis and hypercontractivity.) For families F ⊂ P ([n]) with measure |F|/2 n ∈ [α, 1−α], Theorem 1.3 implies that I[F] ≥ 2α log 2 (1/α). Hence, Friedgut's theorem can be viewed as a structure theorem for families of measure bounded away from 0 and 1, whose total influence lies within a constant multiplicative factor of the minimum possible total influence. Similarly, Friedgut [12], Bourgain [4] and Hatami [19] obtained structure theorems for 'large' subsets of P ([n]) whose 'biased' measure lies within a constant multiplicative factor of the minimum possible, and Kahn and Kalai [20] stated several conjectures on 'small' subsets of P ([n]) satisfying the same condition. The results of [4,12,19] are deep, with many important applications.
In contrast to the results of [4,12,19], which describe the structure of families with total influence within a constant factor of the minimum, our Theorem 1.5 describes the structure of Boolean functions with total influence 'very' close to the minimum. On the other hand, the structure we obtain is very strong -namely, closeness to a genuinely extremal family. Case (1): F is essentially contained in a subcube of codimension 1 (i.e., in a family depending upon just one coordinate), and the total influence of the part of F inside the subcube is 'small'.
Case (2): F is essentially contained in a subcube of codimension 2 (i.e., in a family depending upon just two coordinates), and the total influence of the part of F inside that subcube is 'small'.
Once Proposition 4.1 is established, the main theorem follows by a short induction on n (Proposition 4.1 is needed for the inductive step). It is perhaps fortunate that, for the inductive step, it suffices to pass to subcubes of codimension at most 2. Interestingly, it does not suffice to pass to subcubes of codimension 1, as we explain in Section 4; this might, at first glance, deceive one into abandoning an inductive approach. The proof of Proposition 4.1 is divided into two parts. In the first part, we prove that if |F| is 'sufficiently large' (specifically, if |F| ≥ 2 n−2 (1 + c) for an absolute constant c > 0), then F must satisfy (1); this is the content of Proposition 7.1. In the second part, we prove that if |F| < 2 n−2 (1 + c), then F must satisfy (2); this is the content of Proposition 8.1. The harder part is the first one; the proof is (again) by induction on n, but with six ingredients that are outlined at the beginning of Section 7. Roughly speaking, we define a collection of 'small alterations' which preserve the property of being a counterexample to Proposition 7.1; applying a sequence of these small alterations, we reduce to the case where the family is sufficiently 'well-behaved' for us to successfully apply the inductive hypothesis. (Note that a very similar technique was used in [24].) The second part uses the classical shifting technique [5, 7]: we first reduce to the case where F is monotone increasing; we then choose the coordinate of largest influence (i say), and apply appropriate shifting operators to F to produce a family contained entirely within the codimension-1 subcube {S ⊂ [n] : i ∈ S}; passing to this subcube, we obtain a family of twice the measure of the original family; we then repeat this process until the family is large enough that we can apply Proposition 7.1 (from the first part).
An important component of the proof of Proposition 4.1 is a pair of 'bootstrapping' lemmas showing that if F is 'somewhat' close to being contained in a subcube of codimension 1 or 2, then it must be 'very' close to that subcube. In order to prove these bootstrapping lemmas, we introduce the notion of fractional lexicographic families, as a convenient technical tool. These allow us to analyse how the measure (or 'mass') of a family of small total influence can be distributed between two disjoint codimension-1 subcubes (or between four disjoint codimension-2 subcubes); informally, this distribution cannot differ too much from in the extremal, lexicographically ordered family L.

Organization of the paper
In Section 3, we introduce some definitions and notation, and present some basic facts on influences and shifting. In Section 4, we reduce the main theorem to the intermediate structural result, Proposition 4.1, discussed above. Fractional lexicographic families and their properties are studied in Section 5, and the bootstrapping lemmas are presented in Section 6.
The proof of Proposition 4.1 spans Sections 7-9. The case of 'large' families is covered in Section 7, 'small' families are dealt with in Section 8, and finally we combine these two cases to prove Proposition 4.1 in Section 9. We conclude with some open problems in Section 10.

Notation
We equip P ([n]) with the uniform measure, denoted by µ: We write S ∼ P ([n]) to mean that S is chosen uniformly at randomly from P ([n]).
, and F ⊂ P ([n]), we define the 'sliced' family Note that we view F C B as a subfamily of P ([n] \B), and so µ F B , we say that F and G are weakly isomorphic if there exists an automorphism φ of Q n such that G = φ (F); in this case, we write F ∼ = G. To be completely formal and explicit, if π ∈ Sym([n]) and S ⊂ [n], we write π (S) : For n ∈ N and 2 n µ ∈ {0, 1, . . . , 2 n }, we let L µ,n denote the initial segment of the lexicographic ordering on P ([n]) with measure µ. We write L µ,n for the class of all families weakly isomorphic to L µ,n .
When n is understood, we will write these as L µ and L µ , suppressing the subscript n. We say that a family L ⊂ P ([n]) is lexicographically ordered if it is an initial segment of the lexicographic ordering on P ([n]). We say that a family F ⊂ P ([n]) is monotone increasing (or just increasing) if it is closed under taking supersets, i.e. whenever A ⊂ B ⊂ [n] and A ∈ F, we have B ∈ F.

Influences
Using the notation above, we may define the ith influence of a family F ⊂ P ([n]) by As mentioned in the introduction, we have i.e. the total influence of F is the normalized edge boundary of F. We may therefore restate our main theorem (Theorem 1.5) as follows.
Theorem. There exists an absolute constant C > 0 such that the following holds. Let ε > 0, let F ⊂ P ([n]) be a family of measure µ, and suppose that I [F] ≤ I L µ + ε. Then there exists a family G ⊂ P ([n]) weakly isomorphic to L µ , such that µ (F∆G) ≤ Cε.
(Note that the constant C above is half the constant in the original statement.) It will be more convenient for us to work with the above reformulation. If F ⊂ P([n]) and i ∈ [n], we define the family of i-pivotal sets in F by Note that we have The following lemma will be useful for relating the influence of a family F to the influences of its slices.
The proof is straightforward, and we omit it.

Shifting
The following shifting operator S S,T was introduced by Erdős, Ko and Rado [7] in the case |S| = |T | = 1; for larger values of |S| or |T |, it was introduced by Daykin [5].
These shifting operators are known to be a very useful tool in extremal combinatorics. They were used by Frankl [10] to obtain stability results for the Erdős-Ko-Rado theorem [7], and were recently applied by the authors in [24] to obtain a stability result for the Ahlswede-Khachatrian theorem [1], thus proving a conjecture of Friedgut [13]. A major part of our argument is based on the method of [24].
The following lemma says that if a family F is stable under 'lower-order' shifts, then a shifting operator cannot increase the total influence of F. Proof. Write G = S ST (F). By Lemma 3.1, we have and To prove the claim, it suffices to show that for any family F ⊂ P ([n]) and any i / ∈ S ∪ T , we have The verification of the former assertion is straightforward, and we leave it to the reader.
To prove the latter assertion, we may assume that S ST (F) = F; then We also need the following well-known lemma on the so-called 'monotonization operators' S ∅{i} (see e.g. [21]).

Reduction of Theorem 1.5
In this section we reduce Theorem 1.5 to the following proposition.
Proposition 4.1. There exist absolute constants c 1 , c 2 > 0 such that the following holds. Let 0 < µ ≤ 1 2 , let 0 ≤ ε ≤ c 1 µ, and let F ⊂ P ([n]) be a family with µ (F) = µ and I [F] ≤ I L µ + ε. Then there exists a family G weakly isomorphic to F such that one of the following holds.
Intuitively, Proposition 4.1 says that there is a family G weakly isomorphic to F, such that one of the following holds: either (1) G is essentially contained in the dictatorship D 1 , and the 'essential part' G

{1} {1}
has small total influence, or (2) G is essentially contained in the subcube S {1,2} , and the 'essential part' G  Let G be a family weakly isomorphic to F such that Then there exists π ∈ Sym([n]) and D ⊂ [n] such that G = X D (π(F)). We have It is easy to check that I[F] = 1 + 8t · 2 −t − 24 · 2 −t and I[L µ ] = 1 + 4t · 2 −t − 12 · 2 −t , and therefore if t is sufficiently large depending on c 1 . Observe that and that F is invariant under permuting the coordinates 1 and 2. Hence, if t is large enough that then we may assume that D = / 0 and π = Id, i.e. G can be obtained from F without flipping or permuting any coordinates, so G = F. Hence, We have Finally, we have µ − 1 (F) = 4 · 2 −t . Substituting the latter two facts into (4.1) yields a contradiction.
We now show how to deduce Theorem 1.5 from Proposition 4.1. We state Theorem 1.5 again below (in the influence form), for the convenience of the reader.
Theorem. There exists an absolute constant C > 0 such that the following holds. Let F ⊂ P ([n]) be a family of measure µ (F) = µ, and suppose that I [F] ≤ I L µ + ε. Then there exists a family G ⊂ P ([n]) weakly isomorphic to L µ , such that µ (F∆G) ≤ Cε.
Proof. We prove the theorem by induction on n. If n = 1, then F itself is weakly isomorphic to L µ . Let n ≥ 2, and assume the statement of the theorem holds for smaller values of n.
We now make several reductions. Firstly, we note that the theorem holds for F if and only if it holds for its complement F c , since the complement of a lexicographically ordered family is weakly isomorphic to a lexicographically ordered family. Thus, we may assume w.l.o.g. that µ (F) ≤ 1 2 . Secondly, note that the conclusion of the theorem holds trivially if Cε ≥ 2µ. So we may assume throughout that ε < 2µ C . Provided C ≥ 2/c 1 , we have ε ≤ c 1 µ, so either Case (1) or Case (2) of Proposition 4.1 occurs. First suppose that Case (1) occurs. By replacing F by a family G weakly isomorphic to F if necessary, we may assume that where H is a lexicographically ordered family with respect to the usual ordering 2 ≤ 3 ≤ · · · ≤ n.
Note that The induction hypothesis, and our assumption above on the families H, imply that Rearranging (4.2), we have Putting together (4.3), (4.4) and (4.5), we obtain This completes the proof in Case (1). Suppose now that Case (2) occurs. Replacing F by a family G weakly isomorphic to it, we may assume that where H is a lexicographically ordered family with respect to the usual ordering 3 ≤ · · · ≤ n.
We now have By the induction hypothesis, we have Putting together (4.6), (4.7) and (4.8), we obtain where the last inequality holds provided C ≥ 2 c 2 . This completes the proof.  such that (F ass ) B

Fractional lexicographic families and their properties
[n] is the lexicographically ordered family of measure F(B), for each B ⊂ [n]. If F is a fractional lexicographic family, then by a slight abuse of notation we define µ(F), ) to be the corresponding quantities for the associated family F ass ⊂ P ([m + n]); it is easy to see that these are independent of the choice of m, provided we define Inf i [F ass ] = 0 for all i > n + m.
The usefulness of fractional lexicographic families comes from the fact that inductive arguments enable us to reduce statements about general families to statements about fractional lexicographic families of order n, for small n. Specifically, we need a thorough analysis of the case n = 1 and the case n = 2. These statements encapsulate the idea that families of small total influence can only have their measure split between two disjoint codimension-1 subcubes (or between four disjoint codimension-2 subcubes) in certain ways. The proofs of the statements are technical and the reader is advised (at least at first reading) to read the statements of the lemmas without going into their proofs. Let µ = 2 − j + r, where j ≥ 2 and 0 < r ≤ 2 − j . Observe that The next lemma says roughly that if a fractional lexicographic family L = L µ − ,µ + of order 1 has 0 < µ − ≤ r, then I L µ − ,µ + is somewhat large.
provided c ≤ 1/4. By Lemma 3.1, we have Let L be the fractional lexicographic family of order 2, such that By Lemma 3.1, we have Combining this with (5.18) yields The isoperimetric inequality now implies that completing the proof.

Two 'bootstrapping' lemmas
In this section, we prove two 'bootstrapping' lemmas which say, roughly speaking, that if F is 'somewhat' close to being contained in a subcube of codimension 1 or 2, then it is 'very' close to being contained in that subcube. In what follows, we write µ = 2 − j + r, where j ≥ 2 and 0 < r ≤ 2 − j , we let ε > 0, and we let F ⊂ P ([n]) be a family with measure µ (F) = µ, and with I [F] = I L µ + ε.
Recall that our goal is to prove Proposition 4.1. First, we deal with the case where r is 'large'. In this case, our aim is to show that min{2µ − The following 'bootstrapping' lemma says that this inequality holds provided only that µ − i (F) ≤ r. Lemma 6.1. Let 0 ≤ µ ≤ 1 2 and write µ = 2 − j + r, where j ≥ 2 and r ≤ 2 − j . Let F ⊂ P ([n]) be a family with measure µ (F) = µ, and with I Proof. Using Lemma 3.1, the isoperimetric inequality and Lemma 5.1, we have

Rearranging yields 2µ
We now prove a bootstrapping lemma suitable for the case where r is 'small'. Here, our final goal is to show that there exists a family G weakly isomorphic to F such that either cµ − 1 (G) + 1 2 ε + 1 (G) ≤ ε, or else cµ G\S {1,2} + 1 4 ε ++ 1,2 (G) ≤ ε. We show that one of these inequalities holds provided µ − 1 (G) ≤ µ − 2 (G) ≤ cµ, if c is a sufficiently small positive constant. Lemma 6.2. Let ε > 0, let 0 < µ ≤ 1 2 , and write µ = 2 − j +r, where j ≥ 2 and 0 < r ≤ 2 − j . Let F ⊂ P ([n]) be a family with measure µ (F) = µ, and with Proof. The case where µ − 1 ≤ r is covered by Lemma 6.1, and the case where µ − 2 ≥ 3r can be covered similarly by using the second part of Lemma 5.1 instead of its first part. So we may assume that r ≤ µ − 1 ≤ µ − 2 ≤ 3r. Let L be the fractional lexicographic family of order 2, with Using Lemma 3.1, the isoperimetric inequality and Lemma 5.3, we obtain 7 F is essentially contained in a codimension-1 subcube (µ large) In this section we essentially complete the proof of Proposition 4.1 in the case where µ (F) = 1 4 + r, for r ≤ 1 4 bounded away from 0 (i.e. r ≥ c 1 for some absolute constant c 1 > 0). In this case, by Lemma 6.1 it will suffice to prove the following. where c 2 will be sufficiently small in terms of c 1 . We assume without loss of generality that µ − i ≤ µ + i for each i ∈ [n], and that We also assume that c 1 = 2 −k for some k ∈ N.
The proof of Proposition 7.1 consists of the following six steps, similarly to in [24].
1. We show that there is a 'gap' between 'good' families that satisfy the proposition, and 'bad' families which would furnish a counterexample to it. More precisely, we show that if µ − i (F) > c 1 for each i ∈ [n] and if F 2 ⊂ P ([n]) is a family with µ (F 2 ) = µ (F) and with I [F 2 ] ≤ I [F], which satisfies µ − i (F 2 ) ≤ c 1 for some i ∈ [n], then µ (F∆F 2 ) > c 1 2 . 2. We reduce the proposition to the case where F is increasing.
3. We prove the proposition in the case where F depends on a constant O c 2 (1) number of coordinates. 4. In the other case, where n is large, we show that the 'n-stable' familỹ F := S n,n−1 (S n,n−2 · · · (S n,1 (F))) satisfies µ(F) = µ(F), I[F] ≤ I[F], and µ(F∆F) ≤ c 1 2 . This reduces us to the case where F is an increasing, n-stable family.
5. In the case where F is n-stable and |I n (F)| ≥ 2 (say A = B ∈ I n (F)), we show that if both 6.
Step (5) reduces us to the case where |I n (F)| ≤ 1, i.e. the family F is very evenly balanced in direction n; we can then complete the proof by induction on n.
The next corollary follows immediately from Lemmas 7.2 and 7.3.
If the increasing family G is good, then so is F.
From now on we assume that F is increasing.

Proof in the case where n is small
We now show that Proposition 7.1 holds in the case where n is small. In fact, crudely, we have the following. Proof. If c 2 < 2 −(n−1) , then we must have I [F] = I L µ (note that the influence of any family depending on n variables is of the form i 2 n−1 ). The lemma now follows from the uniqueness part of the isoperimetric inequality.

Reduction to the case where F is n-stable
We say that F is n-stable if S n,i (F) = F for each i ∈ [n − 1], and if A ∪ {n} ∈ F whenever A ∈ F. (As usual, we write S i, j for S {i}{ j} , for brevity.) Here, we show that if F ⊂ P ([n]) is bad and n is large then there exists a c 1 2 -small modification of F that is n-stable. We need the following well-known lemma. We remark that the operator S i, j preserves monotonicity, for each i = j. We also need the following crude upper bound on the total influence of lexicographically ordered families.
Proof. We prove the claim by induction on n. If n = 1 then in fact I L µ ≤ 1. We may assume that µ ≤ 1 2 , since I L 1−µ = I L µ . Hence, the induction hypothesis implies that The following lemma reduces Proposition 7.1 to the case where n is stable. := S n,n−1 (· · · S n,2 (S n,1 (F))) .
Proof. By Lemma 7.2, it suffices to show thatF is a c 1 2 -small modification of F. For this, by Lemma 7.6, it suffices to prove that µ F ∆F ≤ c 1 2 . The key observation is that where i A := min{i : (A ∪ {i}) \ {n} / ∈ F}}, is an injection from F\F to I n (F). Now note that To complete the proof of the lemma, note that (7.4) and (7.5) imply that provided c 2 < 1, using Claim 7.7. Suppose that F is bad. By Lemma 7.5, we may assume that n ≥ 6/c 1 , provided c 2 < 2 −(6/c 1 −1) , and therefore µ(F∆F) ≤ 1 2 c 1 . Hence,F is a c 1 2 -small modification of F, and we are done by Lemma 7.2, provided c 2 ≤ c 1 .
From now on we assume also that F is n-stable.

Proof of Proposition 7.1
We prove the proposition by induction on n. Let c 1 = 2 −k , where k ∈ N. Assume that The case n ≤ k + 1 follows from Lemma 7.5. Let n ≥ k + 2, and let F be as in the hypothesis of the proposition. Suppose for a contradiction that F is bad. By Corollary 7.4, we may assume that F is increasing; by Lemma 7.8, we may assume that F is n-stable, and by Lemmas 7.2, 7.5, and 7.9 we may assume that |I n (F)| ≤ 1. If |I n (F)| = 0, then F does not depend on the nth coordinate and the proposition holds by the induction hypothesis. Suppose that |I n (F)| = 1. Then, by Lemma 3.1, we have Since |F| is odd, and since in the lexicographic ordering, sets containing n alternate with sets not containing n, we have L µ = (L µ − n ,µ + n ) ass , and therefore (7.7) By (7.6) and (7.7), we either have Recall that we are assuming c 1 = 2 −k for some k ∈ N, and that n ≥ k + 2. Since 1/4 + 2 −k ≤ µ = µ − n + 2 −n , 2 n−1 µ − n ∈ Z and n > k, we have µ − n ≥ 1/4 + 2 −k . Moreover, since µ ≤ 1/2, µ + n = µ + 2 −n and 2 n−1 µ + n ∈ Z, we must have µ ≤ 1/2 − 2 −n , and therefore µ + n ≤ 1/2. Hence, Therefore, we may apply the induction hypothesis to one of F We start by reducing to the case where F is increasing.
The key lemma for the proof of Proposition 8.1 is the following.   Proof. To prove (i), first note that F n is increasing, using the fact that S S1 (G) is increasing whenever G is increasing and S S 1 (G) = G for all S ⊂ S with |S | = |S| − 1. Suppose for a contradiction that F n D 1 ; then there exists S ⊂ {2, . . . , n} such that S ∈ F n , so by the monotonicity of F n , we have {2, 3, . . . , n} ∈ F n . But then, by construction of F n , we have D 1 ⊂ F n , and so D 1 ∪ {{2, 3, . . . , n}} ⊂ F n , contradicting the fact that µ(F n ) = µ(F) ≤ 1/2. Statement (ii) follows by repeated application of Lemma 3.3, and (iii) is clear.
The idea of the proof of Proposition 8.1 is as follows. Let F ⊂ P ([n]) be an increasing family as in the hypothesis of the proposition; assume w.l.o.g. that µ − 1 (F) = min i (µ − i (F)). Let F n be the family from Lemma 8.3. By Lemma 8.3, we have {1} ) = 2µ(F). This allows us perform an inductive argument, doubling the measure of the family at each step, and thus reducing to the case of measure somewhat larger than 1/4 (encapsulated in the following lemma, which enables us to do the base case of the induction).  Write µ = 2 − j · µ 0 , where 1 4 + d 1 ≤ µ 0 ≤ 1 2 + 2d 1 and j ∈ N. We prove by induction on j that Proposition 8.1 holds (for increasing families) with the above choice of d. Let j ≥ 1. Suppose w.l.o.g. that F satisifes µ − 1 (F) ≤ µ − 2 (F) ≤ · · · ≤ µ − n (F).
Let F n be as in Lemma 8.3. Let F = (F n )