A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems

We study the thresholds for the property of containing a solution to a linear homogeneous system in random sets. We expand a previous sparse Sz\'emeredi-type result of Schacht to the broadest class of matrices possible. We also provide a shorter proof of a sparse Rado result of Friedgut, R\"odl, Ruci\'nski and Schacht based on a hypergraph container approach due to Nenadov and Steger. Lastly we further extend these results to include some solutions with repeated entries using a notion of non-trivial solutions due to R\'uzsa as well as Ru\'e et al.


Introduction
A k-term arithmetic progression is a set of integers that can be written as {a, a + d, a + (k − 1)d} for some a, d, k ∈ Z, k ≥ 3 and d = 0. The Theorem of van der Waerden [1] states that every finite colouring of [n] = {1, . . . , n} contains a monochromatic k-term arithmetic progression for n large enough. Szemerédi's Theorem [2] strengthened this result by stating that every set of integers with positive natural density contains a k-term arithmetic progression. Rado [3] generalized van der Waerden's result to certain systems of linear equations and Frankl, Graham and Rödl [4] did the same for Szemerédi's extremal result.
A common area of interest is to study 'sparse' or 'random' versions of such results. Consider the binomial random set [n] p where each element in [n] is chosen independently with probability p = p(n). That is [n] p is a random variable sampling from the finite probability space on all subsets of [n] that assigns each T ⊆ [n] the probability P ([n] p = T ) = p |T | (1 − p) n−|T | . Given an integer valued matrix A ∈ M r×m (Z) with r rows and m columns, we largely follow the notation of Schacht [5] and let S(A) = {x ∈ Z m : A · x T = 0 T } be the set of all solutions and S 0 (A) = {x = (x 1 , . . . , x m ) ∈ S(A) : x i = x j for i = j} the set of all proper solutions. We call A irredundant if S 0 (A) = ∅. Given a set of integers T and s ∈ N we write if for every finite partition T 1∪ . . .∪ T s = T there exists 1 ≤ i ≤ s such that T i ∩ S 0 (A) = ∅. An irredundant matrix A ∈ M r×m (Z) is partition regular if for every s ∈ N we have [n] → s A for n large enough. Rado [3] gave the column condition as a characterization of partition regular matrices. Rödl and Ruciński [6] formulated a sparse version of Rado's Theorem that was later completed by Friedgut, Rödl and Schacht [7]. To state it let ∅ = Q ⊆ [m] be any set of column indices and define r Q = rk(A) − rk(A Q ) where A Q is the matrix obtained by keeping only the columns which are indexed by Q and the rank of a matrix A is denoted as rk(A). Here A ∅ is the empty matrix with rk(A ∅ ) = 0. The maximum 1-density of a given matrix A ∈ M r×m (Z) is defined as We will later see that for the specific kinds of matrices under consideration, this is indeed well-defined, that is |Q| − r Q − 1 > 0 for all Q ⊆ [m] satisfying |Q| ≥ 2. This allows us to state the following sparse version of Rado's Theorem.
The current proof of this theorem is quite involved. A first goal of this note is to provide a short proof of the 1-statement in Theorem 1.1 following the ideas of Nenadov and Steger's short proof of a sparse Ramsey Theorem [8]. This approach combines the recently developed hypergraph container framework by Balogh, Morris and Samotij [9] as well as Saxton and Thomason [10] with a supersaturation result of Frankl, Graham and Rödl [4].
Schacht [5] as well as independently Conlon and Gowers [11] also stated a sparse version of Szémeredi's Theorem. Schacht also extended it to density regular systems as well as Schur triples. Given a set of integers T and ǫ > 0 we write is density regular if for all ǫ > 0 we have [n] → ǫ A for n large enough. Frankl, Graham and Rödl [4] characterized density regular systems as those that are invariant, that is they satisfy A·1 = 0. Lastly, we say that a matrix A ∈ M r×m (Z) is positive if S(A) ∩ N m = ∅ and abundant if any r × (m − 2) submatrix obtained from A by deleting two columns has the same rank as A. Every density regular system is clearly partition regular and partition regular systems are irredundant and positive by definition. We will later see in Lemma 2.2 that they are also abundant. The second goal of this note is to extend Schacht's statement to the broadest group of matrices possible. To prove this generalization we derive a supersaturation result from a removal lemma due to Král', Serra and Vena [12] and combine it with a corollary of the hypergraph containers due to Balogh, Morris and Samotij [9].
Given some matrix A ∈ M r×m (Z) let ex(n, A) be the size of the largest subset of [n] not containing a proper solution and define π(A) = lim n→∞ ex(n, A)/n. Observe the clear parallels to the Turán number of a graph. Clearly density regular systems satisfy π(A) = 0 and for other systems systems we have π(A) > 0. One can easily bound this value away from 1, as we will later see in Lemma 2.1. These definitions and observations allow us to state the following sparse extremal result. Theorem 1.2. For every ǫ > π(A), r, m ∈ N such that m ≥ 3 and matrix A ∈ M r×m (Z) there exist constants c = c(A, ǫ) and C = C(A, ǫ) such that the following holds. If A is irredundant, positive and abundant then For singe-line equations it is common in combinatorial number theory to not just limit oneself to proper solutions, that is solutions with no repeated entries, but to also consider certain non-trivial solutions which may have some repeated entries. Ruzsa [13] gave a definition for non-trivial solutions in this scenario and more recently Rué, S. and Zumalacárregui [14] extended it to include arbitrary homogeneous linear systems of equations. A third and final goal will therefore be to extend the previously stated sparse results to include non-trivial solutions. We need to introduce some notation in order to give a formal definition of this notion.
Given a solution x = (x 1 , . . . , x m ) ∈ S(A) let A solution x ∈ S(A) is now defined to be non-trivial if rk(A p(x) ) = rk(A). We denote the set of all nontrivial solutions by Lastly, given a set of integers T , an integer s ∈ N and some ǫ > 0, we write if for every finite partition T 1∪ . .
if every subset S for which |S|/|T | ≥ ǫ also satisfies S ∩ S 1 (A) = ∅. This is just a direct extension of the previous notation to include non-trivial solutions. We now have the following two statements.
Observe that these results are stronger than the 0-statements of Theorem 1.1 and Theorem 1.2 and that their respective 1-statements supply matching counter-statements to Theorem 1.3 and Theorem 1.4. We will therefore only require proofs of the 1-statements of Theorem 1.1 and Theorem 1.2 as well as the 0-statements in Theorem 1.3 and Theorem 1.4.
Outline. In the remainder of this note we will first state some preliminaries in Section 2 about linear systems of equations, their subsystems and supersaturation results as well as introduce hypergraph containers. We will then proceed by providing short proofs based on hypergraph container results of the 1-statements in Theorem 1.2 and Theorem 1.1 in Sections 3 and 4 respectively. The 0-statements of Theorem 1.3 and Theorem 1.4 will be proven in Sections 5 and 6 respectively.
Acknowledgements. I would like to thank Lluís Vena for his tremendous help and input regarding the removal lemma and the proof of the supersaturation results. I would also like to thank Juanjo Rué for his general assistance and supervision.

Preliminaries
Given some matrix A ∈ M r×m (Z), we have previously defined ex(n, A) to be the size of the largest subset of [n] not containing a proper solution and π(A) = lim n→∞ ex(n, A)/n. Erdős and Turan [15] determined that the size of the largest Sidon set, that is A = ( 1 1 −1 −1 ), satisfies ex(n, A) = Θ( √ n). For sum-free subsets, that is A = ( 1 1 −1 ), it is also easy to see that π(A) = 1/2. Hancock and Treglown [16] very recently extended this to matrices of the form A = ( p q −r ) where p, q, r ∈ N such that p ≥ q ≥ r. Unfortunately, unlike the Erdős-Stone-Simonovits Theorem [17,18] in the graph case, no exact characterization of π(A) is known for arbitrary matrices A. However, the following lemma shows that one can still easily bound this value away from 1 for every irredundant and positive matrix. Proof. Let x = (x 1 , . . . , x m ) ∈ S 0 (A) ∩ N m . Clearly we also have j · x = (jx 1 , . . . , jx m ) ∈ S 0 (A) ∩ N m for any j ≥ 1. Now for n ≥ m max i (x i ) we observe that every i ∈ [n] can appear in at most m of the J = ⌊n/ max i (x i )⌋ solutions x, 2 · x, . . . , J · x ∈ [n] m , so every subset of [n] that avoids S 0 (A) is missing at least J/m elements. It follows that π(A) ≤ (n − J/m)/n ≤ 1 − 1/(m max i (x i )) < 1.
Partition and density regular matrices are irredundant and positive by definition. The next statement shows that Theorem 1.2 does indeed extend already existing results by proving that they are also abundant.
Lemma 2.2. If a given A ∈ M r×m (Z) is partition or density regular, then it is abundant.
Proof. Rado characterized partition regular matrices as those that satisfy the column condition, that is it is possible to re-order the column vectors c 1 , . . . , c m of A, so that for some choice of indices 0 = m 0 < m 1 < · · · < m t = m setting b i = mi j=mi−1+1 c j for i = 1, . . . , t gives b 1 = 0 and b i can be expressed as a as a rational linear combination of c 1 , . . . , c mi−1 for all i ∈ {2, . . . , t}.
Assume now that A is non-abundant, that is there exists a submatrix obtained by omitting two columns that has rank strictly smaller than rk(A). It follows that through basic row operations A can be transformed into a matrix of full rank whose last row contains only two non-zero entries a, b ∈ Z\{0}. As the matrix is partition regular, it is also irredundant and hence a = −b, that is a + b = 0. It follows that in order to satisfy the first requirement of the column condition, the columns need to be arranged such that there is a 0 in the last entry of the first column. However, there now must exist some 2 ≤ i ≤ t such that the last entry in b i is non-zero while the last entries in b 1 , . . . , b i−1 are zero, violating the second requirement of the column condition. It follows that A must have been abundant.
Let us consider some examples to illustrate these categories. A = ( 1 1 −2 ), that is the matrix associated with 3-term arithmetic progression, is density regular by Roth's Theorem [19]. It therefore is trivially also partition regular, which was previously established by van der Waerden [1]. The matrix associated with k-term arithmetic progressions is density regular and therefore abundant by Szémeredi's Theorem [2]. A = ( 1 1 −1 ) is not density regular, but by Schur's Theorem it is still partition regular. Lastly, A = ( 1 1 −r ) for r ∈ N\{1, 2} is neither partition nor density regular but it is abundant. For some more examples see [14].

Counting Solutions
We extend the notation from the introduction to inhomogeneous systems of linear equations. Given some matrix A ∈ M r×m (Z) and column vector b ∈ Z r we write . We remark that by elementary properties of systems of linear equations, we have the trivial upper bound The next lemma is due to Janson and Ruciński [20] and establishes a lower bound that matches this up to a constant for homogeneous systems. It could be trivially extended to include non-homogeneous systems. [20]). Let r, m ∈ N and a matrix A ∈ M r×m (Z) be given. If S 0 (A) ∩ N m is non-empty then there exists a constant c 0 = c 0 (A) > 0 such that

Lemma 2.3 (Janson and Ruciński
In order to determine the exact asymptotic value of | S(A)∩[n] m |/n m−rk(A) or | S 0 (A)∩[n] m |/n m−rk(A) , one needs to employ Ehrhart's Theory, see for example Rué et al. [14].
Lastly, let P (A) = {p(x) : x ∈ S 1 (A)} denote the family of all set partitions of the column indices [m] stemming from non-trivial solutions. The following lemma gives us the necessary tool to handle non-trivial solutions with repeated entries.
. , x min(TS ) ) ∈ T |p| as well as x Q ∈ S(A p ) as can be readily seen by the definition of A p . Since p = p(x), the vector x Q would furthermore be proper, so that An easy corollary of this result is clearly that if there are no proper solutions to the system A p in a set, then there can also not be non-trivial solutions to A whose repetitions are indicated by p.

Subsystems
The notion of subsystems was originally introduced by Rödl and Ruciński [6] when developing a sparse version of Rado's Partition Theorem. Recall the definitions from the introduction, especially r Q = rk(A) − rk(A Q ). Observe that we can without loss of generality assume that A is of full rank for this part, since the solution space is unaffected by this assumption. This will simplify notation significantly.
For a given matrix A ∈ M r×m (Z) and column indices ∅ ⊆ Q ⊆ [m], we will now construct through basic row operations a matrix that tries to encapsulate the information contained in A through the columns indexed by Q. Denote the rows of A by a 1 , a 2 , . . . , a r so that the rows of A Q and A Q are respectively a Q 1 , a Q 2 , . . . , a Q r and a Q 1 , a Q 2 , . . . , a Q r . Here we allow for empty vectors and matrices. If rk(A Q ) < rk(A), then we can express exactly r Q > 0 of the r rows of A Q as linear combinations of the rest, that is there are indices i 1 < · · · < i rQ ∈ [m] and integers d i , d j i ∈ Z for i ∈ {i 1 , . . . , i rQ } and j ∈ [m]\{i 1 , . . . , i rQ } so that Consider now the following integer-valued matrix with r Q rows and |Q| columns To illustrate this construction further, note that if we assume that the column indices are appropriately ordered, that is Q = {1, . . . , |Q|}, then the matrix A (without the assumption of being of full rank) can be rewritten as through elementary row operations, that is A = P −1 1 · B where P 1 ∈ M r×r is an invertible rectangular matrix. We have S(B) = S(A), that is the homogeneous solution space remains unchanged, at least up to the column permutation necessary to ensure that Q = {1, . . . , |Q|}.
Observe that the matrix A[Q] is only well defined up to our choices of indices i k and coefficients d i , d j i . However, the homogeneous solution space S(A[Q]) is independent of these, so we pick one representative for each ∅ ⊆ Q ⊆ [m] and refer to it as the subsystem of A induced by Q. The notation A[Q] will refer to this particular representative. We state the following simple observations, that are immediately clear by considering Equation (14).
trivially also fulfils these properties for any Q ⊆ [m] such that r Q > 0.
The following lemma now establishes some results regarding the rank of subsystems of abundant matrices. It also verifies that the maximum 1-density parameter given in the introduction is indeed well-defined for abundant matrices. The next lemma is crucial and establishes that a lack of non-trivial solutions to a subsystem of A also implies a lack of non-trivial solutions to the full system. A proof of this as well as the previous statement can be found in Kusch et al. [21]. Note that this was previously proven by Rödl and Ruciński for proper solutions [6]. We end our observations about subsystem by stating the following easy proposition. It covers some trivial cases not considered by Theorem 1.4 and will in fact be needed later in the proof of it. p | = o(np/2) and therefore for any given set of positive density, we can remove one element per solution and still asymptotically almost surely have a solution-free set of that same density. This proves the desired result.

Removal Lemma and Supersaturation Results
A common ingredient to proving sparse results are robust versions of the deterministic statement, referred to as supersaturation results. In the graph setting such a result is folklore and easy to prove. A number theoretical counterpart is Varnavides [22] robust version of Szemerédi's Theorem which states that a set of positive density contains not just one, but a positive proportion of all k-term arithmetic progressions. Frankl, Graham and Rödl [4] formulated such results for both for partition and density regular systems.  We will extend Lemma 2.10 to cover the scope of this note by using an arithmetic removal lemma. Green [23] first formulated such a statement for linear equations in an abelian group. Later Shapira [24] as well as independently Král', Serra and Vena [12] proved a removal lemma for linear maps in finite fields. We will state it here in a simplified version.
Theorem 2.11 (Removal Lemma [12]). Let F q be the finite field of order q. Let X ⊂ F q be a subset of F q and A ∈ M r×m (F q ) a matrix of full rank. For S = {x ∈ F m q : A · x T = 0 T } and every ǫ > 0 there exists an η = η(ǫ, r, m) such that if |S ∩ X m | < η |S| then there exists a set X ′ ⊂ X with |X ′ | < ǫq and S ∩ (X\X ′ ) m = ∅.
Applying this result, we formulate the following extension of Lemma 2.10. Proof. Let q = q(A, n) be a prime number between 2mn max(|A|) and 4mn max(|A|) and F q the finite field with q elements. Here max(|A|) refers to the maximal absolute entry in A. Note that such a prime number exists for example because of the Bertrand-Chebyshev Theorem. We have F q ∼ = Z q and we can identify the integers with their corresponding residue classes in F q . The matrix A now defines a map from F m q to F r q . A solution in S(A) clearly lies in the S and, as we have chosen q large enough, all canonical representatives from S ∩ [n] m also lie in S(A) ∩ [n] m for n ≥ max |A|.
Next, set δ ′ = (δ + π(A))/2 and let n be large enough such that any subset of density at least δ ′ in [n] contains a proper solutions. Note that δ > δ ′ > π(A). Given a subset T ⊆ [n] satisfying |T | ≥ δn consider the corresponding set X of residue classes in F q . One needs to remove at least (δ − δ ′ )n elements from T in order for T m to avoid S 0 (A) in [n], so one needs to remove at least an proportion of elements in F q from X so that X m avoids S in F q . It follows from Theorem 2.11 that |S ∩ X m | ≥ η|S| for some η = η(ǫ, r, m). Since we have chosen q large enough, it follows that T contains at least an η proportion of S(A) ∩ [n] m . An easy consequence of Equation (9)

Hypergraph Containers
The development of hypergraph containers by Balogh, Morris and Samotij [9] as well as independently Thomason and Saxton [10] has opened a new, easy and unified framework to proving sparse results. Let us start by stating the Hypergraph Container Theorem as given by Balogh, Morris and Samotij. Given a hypergraph H we denote its vertex set by V (H) and its set of hyperedges by E(H). The cardinality of these sets will be respectively denoted by The statement gives the existence of a small number of containers F and some fingerprints T so that every independent set I in H is identified with a fingerprint g(I) that determines a container f (g(I)) which contains the independent set.
Next, let H = (H n ) n∈N be a sequence of m-uniform hypergraphs and let α ∈ [0, 1). We say that H is α-dense if for every δ > 0, there exist some ǫ > 0 such that for U ⊆ V (H n ) which satisfies |U | > (α + δ) v(H n ) we have e(H n [U ]) > ǫ e(H n ) for n large enough. Balogh, Morris and Samotij proved the following consequence of their container statement.
Theorem 2.14 (Sparse Sets through Hypergraph Containers, Theorem 5.2 in [9]). Let H = (H n ) n∈N be a sequence of m-uniform hypergraphs, α ∈ [0, 1) and let C > 0. Suppose that q = q(n) is a sequence of probabilities such that for all sufficiently large n and for every ℓ ∈ {1, . . . , m} we have If H is α-dense, then for every δ > 0, there exists a constant c = c(C, α, m) > 0 such that if p(n) > c q(n) and p(n) v(H n ) → ∞ as n → ∞, then asymptotically almost surely We will make use of this statement in order to obtain a proof for the 1-statement of Theorem 1.2. For a proof of the 1-statement of Theorem 1.1 such a ready-made statement does not exist and we will follow Nenadov and Steger's [8] short proof of a sparse Ramsey statement by applying Theorem 2.13. Observe that H n can be a multigraph, that is multiple edges are allowed, but the multiplicity of each edge is clearly bounded by m!. We do this to simplify counting, since this way we have |E(H n )| = | S 0 (A)∩[n] m |. We observe that we can limit ourselves to proper solutions when proving the 1-statement. Corollary 2.12 now states that H = (H n ) n∈N is π(A)-dense. In order to apply Theorem 2.14, it remains to determine a sequence q = q(n) satisfying the required condition. The following lemma gives us upper bounds for the maximum ℓ-degrees in H n . Proof. For H = (H n ) as defined above and ℓ ∈ {1, . . . , m} we have where S(ℓ) denotes the set of permutations of ℓ elements. We have also made extensive use of the notation defined in the introduction as well as as the trivial upper bound for the number of solutions stated in Equation (9).
Note that r Q = 0 for any Q ⊆ [m] satisfying |Q| = 1 due to Lemma 2.6 and that there exists c 0 = c 0 (A) > 0 such that e(H n ) ≥ c 0 n m−rk(A) due to Lemma 2.3. Using Lemma 3.1 we now observe that For ℓ ∈ {2, . . . , m} we again apply Lemma 3.1 to see that Lastly we observe that n −1/m1(A) v(H n ) = n 1−1/m1(A) → ∞ as m 1 (A) > 1. It follows that the prerequisites of Theorem 2.14 hold for C = (m! m m )/c 0 , q = q(n) = n −1/m1(A) and we can choose the c = c(A, ǫ) in Theorem 1.2 to be equal to the c = c(C, π(A), m) as given by Theorem 2.14.

A Short Proof of the 1-statement in Theorem 1.1
As stated in the introduction, this result was previously proven by Friedgut, Rödl and Schacht [7] as well as independently Conlon and Gowers [11]. This prove merely serves as a short version that follows the short proof of a sparse Ramsey result due to Nenadov and Steger [8].
We will need two ingredients in order to prove the 1-statement of Theorem 1.1. The first will be the following easy corollary to Lemma 2.9. The second ingredient is stated in the following corollary that is obtained by applying the Hypergraph Container Theorem to the hyperpgraph of solutions.
Furthermore, for every set T ⊆ [n] satisfying S 0 (A) ∩ T m = ∅ there exists 1 ≤ i ≤ t such that Proof. Let H n again be the hypergraph with vertex set V (H n ) = [n] and edge multiset We have previously observed that there exists a c > 0 such that ∆ ℓ ≤ c p(n) ℓ−1 e(H n )/v(H n ) for p = p(n) = n −1/m1(A) . We observe that H n is trivially Applying Theorem 2.13 gives the desired statement.
We are now ready to give a short proof of the 1-statement in Theorem 1.1 following the ideas of Nenadov and Steger [8]. Let ǫ, δ > 0 be as in Corollary 4.1 and let t = t(n), c 0 , S 1 , . . . , S t and C 1 , . . . , C t be as in Observe that the two events S j1 , . . . , S js ⊆ [n] p and [n]\(C j1 , . . . , C js ) ∩ [n] p = ∅ are independent, so that we have We bound this by choosing k = | s j=1 S j | ≤ sc 0 n 1−1/m1(A) , then picking k elements and lastly deciding for each element in this selection in which of the S i it is contained, so that we have Lastly we note that for c > 0 the function f (x) = (c/x) x is increasing for 0 ≤ x ≤ c/e since d/dx f (x) = (c/x) x (log(c/x) − 1). We have chosen C large enough so that for n large enough we have As desired it follows that [n] p → s A asymptotically almost surely for p ≥ C n −1/m1(A) .
6 Proof of Theorem 1.4 -A Szémeredi-type 0-statement Due to Lemma 2.4 we know that We will therefore analyze the individual probabilities P [n] p → ǫ A p for each p ∈ P (A). The constant c = c(A, ǫ) will be define later in Equation (27). We start by first stating the following proposition, which restricts the statement of Theorem 1.4 to proper solutions. Its proof will be given at the end of this section. For |p| < m we now observe that A p again clearly is irredundant and positive since p indicates the repeated entries of an actual solution in S 1 (A). If A p is not abundant, then Proposition 2.8 states that lim n→∞ P [n] p → ǫ A p = 0 for p = p(n) ≤ c n −1/m1(A) = o(1) independent of the constant c. If A p is abundant, then we can apply Proposition 6.1 to it. If we assume as in the proof of Theorem 1.3 that m 1 (A) = (m − 1)/(m − rk(A) − 1), then we again have n −1/m1(A) = o n −1/m1(Ap) and therefore lim n→∞ P [n] p → s A p = 0 for p = p(n) ≤ c n −1/m1(A) independent of c. Lastly, let |p| = m, that is p = {{1}, . . . , {m}} and therefore A p = A. Proposition 6.1 applies to A and therefore we obtain the desired statement with c = c(A, ǫ) as given by Proposition 6.1. The desired statement now follows due to Equation (22).
Proof of Proposition 6.1. Observe that the expected number of elements in [n] p is We also note that due to Lemma 2.3 there exists c 0 = c 0 (A[Q]) such that and due to Equation (9) we also have for b ∈ Z r and ∅ = Q ⊆ [m] that Following the alteration method as used for example by Schacht [5], we make three case distinctions. For this, we define the maximum density of A to be Note the difference to the previously defined maximum 1-density. The constant c = c(A, ǫ) will be stated later in context in Equation (27). Note that we need to cover the whole range of 0 ≤ p(n) ≤ c n −1/m1(A) since we are not dealing with a monotone property. Markov's Inequality and Lemma 2.7 therefore give us lim n→∞ P | S 0 (A) ∩ [n] m p | = 0 = 0, see also Rué et al. [14]. It clearly follows that we also have lim n→∞ P ([n] p → ǫ A) = 0 for any ǫ > 0 if p = p(n) ≪ n −1/m(A) .
We observe that due to Equation (24) and the assumption on m 1 (A) we now have Var(X n ) = o(E(X) 2 ).
Chebyshev's inequality therefore gives us P (|X − E(X) | ≥ E(X)) = o(1) so that S 0 (A) ∩ [n] m p ≤ 2 E(X) = (1 − ǫ) np/2 asymptotically almost surely. It follows that, given a set of density ǫ, we can remove one element from [n] p for each solution in S 0 (A) ∩ [n] m p and asymptotically almost surely still be left with a set of density ǫ, so that lim n→∞ P ([n] p → ǫ A) = 0 for any ǫ > 0 if n −1/m(A) ≪ p = p(n) ≤ c n −1/m1(A) where c = c(A, ǫ) as given in Equation (27).