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

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by  Christoph Spiegel

A Note on Sparse Supersaturation and Extremal Results
for Linear Homogeneous Systems
Christoph Spiegel 1
Abstract
7 Westudythethresholdsforthepropertyofcontainingasolutiontoalinearhomogeneoussystem
1 inrandomsets. WeexpandaprevioussparseSz´emeredi-typeresult ofSchachttothebroadest class
0 of matrices possible. We also provide a shorter proof of a sparse Rado result of Friedgut, Ro¨dl,
2
Rucin´skiandSchachtbasedonahypergraphcontainerapproachduetoNenadovandSteger. Lastly

n we further extend these results to include some solutions with repeated entries using a notion of
a non-trivialsolutions dueto Ru´zsa as well as Ru´eet al.
J

6
1 Introduction

]
O
Ak-term arithmetic progression isasetofintegersthatcanbewrittenas a,a+d,a+(k 1)d forsome
C { − }
a,d,k Z, k 3 and d = 0. The Theorem of van der Waerden [1] states that every finite colouring of
. ∈ ≥ 6
h [n]= 1,...,n containsamonochromatick-termarithmeticprogressionfornlargeenough. Szemer´edi’s
t { }
a Theorem [2] strengthened this result by stating that every set of integers with positive natural density
m contains a k-term arithmetic progression. Rado [3] generalized van der Waerden’s result to certain
[ systems of linear equations and Frankl, Graham and Ro¨dl [4] did the same for Szemer´edi’s extremal

result.
1
v A common area of interest is to study ’sparse’ or ’random’ versions of such results. Consider the
1 binomial random set [n] where each element in [n] is chosen independently with probability p = p(n).
p
3
That is [n] is a random variable sampling from the finite probability space on all subsets of [n] that
6 p
1 assigns each T [n] the probability P([n]p =T) = p|T|(1 p)n−|T|. Given an integer valued matrix
⊆ −
0 A (Z) with r rows and m columns, we largely follow the notation of Schacht [5] and let
r×m
. ∈ M
1 (A)= x Zm :A xT =0T be the set of all solutions and (A)= x=(x ,...,x ) (A):x =
0 1 m i
0 Sx for i={ j∈the set·of all prop}er solutions. We call A irredundSant if {(A)= . Given a s∈etSof integer6 s
j 0
7 6 } S 6 ∅
T and s N we write
1
∈
: T A (1)
v →s
i if for every finite partition T ˙ ... ˙ T = T there exists 1 i s such that T (A) = . An
X 1 s i 0
∪ ∪ ≤ ≤ ∩S 6 ∅
irredundant matrix A (Z) is partition regular if for every s N we have [n] A for n large
r ∈ Mr×m ∈ →s
a enough. Rado [3] gave the column condition as a characterizationof partition regular matrices.
Ro¨dl and Rucin´ski [6] formulated a sparse version of Rado’s Theorem that was later completed by
Friedgut, Ro¨dl and Schacht [7]. To state it let = Q [m] be any set of column indices and define
∅ 6 ⊆
r =rk(A) rk(AQ) where AQ is the matrix obtained by keeping only the columns which are indexed
Q
−
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 (Z) is defined as
r×m
∈M
Q 1
m (A)= max | |− . (2)
1
Q⊆[m] Q rQ 1
| |− −
2≤|Q|
1Universitat Polit`ecnica de Catalunya and Barcelona Graduate School of Mathematics, Department of Mathematics,
Edificio Omega, 08034 Barcelona, Spain. E-mail: [email protected]. Supported by the Spanish Ministerio de
Econom´ıayCompetitividadFPIgrantundertheprojectMTM2014-54745-P.
1
We will later see that for the specific kinds of matrices under consideration, this is indeed well-defined,
that is Q r 1 > 0 for all Q [m] satisfying Q 2. This allows us to state the following sparse
Q
| |− − ⊆ | |≥
version of Rado’s Theorem.
Theorem 1.1 (R¨odl and Rucin´ski [6], Friedgut, Ro¨dl and Schacht [7]). For every r,m,s N and
∈
partition regular matrix A (Z) there exist constants c=c(A,s) and C =C(A,s) such that
r×m
∈M
0 if p(n) cn−1/m1(A),
nl→im∞P([n]p →s A)= 1 if p(n)≤Cn−1/m1(A).
(cid:26) ≥
Thecurrentproofofthistheoremisquiteinvolved. Afirstgoalofthisnoteistoprovideashortproof
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,MorrisandSamotij [9]as well as SaxtonandThomason[10] with a supersaturationresult of
Frankl, Graham and Ro¨dl [4].
Schacht[5]aswellasindependentlyConlonandGowers[11]alsostatedasparseversionofSz´emeredi’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
T A (3)
ǫ
→
if every subset S for which S /T ǫ satisfies S (A) = . An irredundant matrix A (Z)
0 r×m
| | | | ≥ ∩S 6 ∅ ∈ M
is density regular if for all ǫ > 0 we have [n] A for n large enough. Frankl, Graham and Ro¨dl [4]
ǫ
→
characterizeddensityregularsystemsasthosethatareinvariant,thatistheysatisfyA 1=0. Lastly,we
·
saythatamatrixA (Z) ispositive if (A) Nm = andabundant ifanyr (m 2)submatrix
r×m
∈M S ∩ 6 ∅ × −
obtainedfromAbydeletingtwocolumnshasthesamerankasA. Everydensityregularsystemisclearly
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
Kr´al’, Serra and Vena [12] and combine it with a corollary of the hypergraph containers due to Balogh,
Morris and Samotij [9].
GivensomematrixA (Z)letex(n,A)bethesizeofthelargestsubsetof[n]notcontaininga
r×m
∈M
proper solution and define π(A)=lim ex(n,A)/n. Observe the clear parallels to the Tur´an number
n→∞
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 (Z) there exist
r×m
∈ ≥ ∈ M
constants c = c(A,ǫ) and C = C(A,ǫ) such that the following holds. If A is irredundant, positive and
abundant then
0 if p(n) cn−1/m1(A),
lim P([n] A)= ≤
n→∞ p →ǫ 1 if p(n) Cn−1/m1(A).
(cid:26) ≥
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 Ru´e, S. and Zumalac´arregui [14] extended it to include arbitrary
homogeneouslinearsystemsofequations. Athirdandfinalgoalwillthereforebetoextendthepreviously
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 ,...,x ) (A) let
1 m
∈S
p(x)= 1 j m:x =x :1 i m (4)
i j
{ ≤ ≤ } ≤ ≤
(cid:8) (cid:9)
2
denote the set partition of the column indices [m] indicating the repeated entries in x. Note that for
x (A) we have p(x) = 1 ,..., m . Given some set partition p of 1,...,m , let A denote the
0 p
∈ S {{ } { }} { }
matrix obtained by summing up the columns of A according to p, that is for p = T ,...,T such that
1 s
{ }
min(T ) < < min(T ) for some 1 s m and c the i-th column vector of A for every i [m], we
1 s i
··· ≤ ≤ ∈
have
Ap = ci ci ... ci . (5)
(cid:18) i∈T1 (cid:12) i∈T2 (cid:12) (cid:12) i∈Ts (cid:19)
A solution x (A) is now defined tPo be no(cid:12)n-trPivial if(cid:12)rk(A )(cid:12) =Prk(A). We denote the set of all non-
(cid:12) (cid:12) p(x)(cid:12)
∈S
trivial solutions by (A) = x (A) : rk(A ) = rk(A) , that is we have (A) (A) (A).
1 p(x) 1 0
S { ∈ S } S ⊇ S ⊇ S
Lastly, given a set of integers T, an integer s N and some ǫ>0, we write
∈
T ⋆ A (6)
→s
if for every finite partition T ˙ ... ˙ T =T there exists 1 i s such that T (A)= and
1 s i 1
∪ ∪ ≤ ≤ ∩S 6 ∅
T ⋆ A (7)
→ǫ
if every subset S for which S /T ǫ also satisfies S (A)= . This is just a direct extension of the
1
| | | |≥ ∩S 6 ∅
previous notation to include non-trivial solutions. We now have the following two statements.
Theorem 1.3. For every r,m,s N and partition regular matrix A (Z) there exists a constant
r×m
∈ ∈M
c=c(A,s) such that
lim P([n] ⋆ A)=0 if p(n) cn−1/m1(A).
n→∞ p →s ≤
Theorem 1.4. For every ǫ > 0, r,m N such that m 3 and irredundant, positive and abundant
∈ ≥
matrix A (Z) there exists a constant c=c(A,ǫ) such that
r×m
∈M
lim P([n] ⋆ A)=0 if p(n) cn−1/m1(A).
n→∞ p →ǫ ≤
Observe that these results are stronger than the 0-statements of Theorem 1.1 and Theorem 1.2 and
thattheirrespective1-statementssupplymatchingcounter-statementstoTheorem1.3andTheorem1.4.
We will thereforeonly require proofs ofthe 1-statements ofTheorem1.1and Theorem1.2as wellas 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 Theorem1.2 and Theorem1.1 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. IwouldliketothankLlu´ısVenaforhistremendoushelpandinputregardingthe
removal lemma and the proof of the supersaturation results. I would also like to thank Juanjo Ru´e for
his general assistance and supervision.
2 Preliminaries
GivensomematrixA (Z),wehavepreviouslydefinedex(n,A)tobethesizeofthelargestsubset
r×m
∈M
of[n]notcontainingapropersolutionandπ(A)=lim ex(n,A)/n. Erdo˝sandTuran[15]determined
n→∞
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 Erdo˝s-Stone-Simonovits Theorem [17, 18] in the graph
≥ ≥
case,noexactcharacterizationofπ(A)isknownforarbitrarymatricesA. However,thefollowinglemma
shows that one can still easily bound this value away from 1 for every irredundant and positive matrix.
3
Lemma 2.1 (Folklore). Every irredundant and positive matrix A (Z) satisfies π(A)<1.
r×m
∈M
Proof. Let x = (x ,...,x ) (A) Nm. Clearly we also have j x = (jx ,...,jx ) (A) Nm
1 m 0 1 m 0
∈ S ∩ · ∈ S ∩
for any j 1. Now for n mmax (x ) we observe that every i [n] can appear in at most m of the
i i
≥ ≥ ∈
J = n/max (x ) solutions x,2 x,...,J x [n]m, so every subset of [n] that avoids (A) is missing
i i 0
⌊ ⌋ · · ∈ S
at least J/m elements. It follows that π(A) (n J/m)/n 1 1/(mmax (x ))<1.
i i
≤ − ≤ −
Partition and density regular matrices are irredundant and positive by definition. The next state-
ment 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 (Z) is partition or density regular, then it is abundant.
r×m
∈M
Proof. Rado characterized partition regular matrices as those that satisfy the column condition, that is
it is possible to re-orderthe columnvectorsc ,...,c ofA, so that for some choiceof indices 0=m <
1 m 0
m < <m =m setting b = mi c for i=1,...,t gives b =0 and b can be expressed as
1 ··· t i j=mi−1+1 j 1 i
a as a rational linear combination of c ,...,c for all i 2,...,t .
P 1 mi−1 ∈{ }
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 transformedintoamatrixoffullrankwhoselastrowcontainsonly twonon-zeroentriesa,b Z 0 .
∈ \{ }
As the matrix is partition regular, it is also irredundant and hence a= b, that is a+b=0. It follows
6 − 6
that in order to satisfy the first requirement of the column condition, the columns need to be arranged
suchthatthereisa0inthe lastentryofthe firstcolumn. However,therenowmustexistsome2 i t
≤ ≤
such that the last entry in b is non-zero while the last entries in b ,...,b are zero, violating the
i 1 i−1
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
1 2 1
−
1 2 1
A= − ... ∈M(k−2)×k (8)
 
 1 2 1 
 − 
 
is density regularandtherefore abundantby Sz´emeredi’sTheorem[2]. A=( 1 1 1 )is notdensity
−
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].
2.1 Counting Solutions We extend the notation from the introduction to inhomogeneoussystems of
linear equations. Given some matrix A (Z) and column vector b Zr we write (A,b)= x
r×m
∈M ∈ S { ∈
Zm : A xT = bT , (A,b) = x = (x ,...,x ) (A,b) : x = x for i = j and (A,b) = x
0 1 m i j 1
· } S { ∈ S 6 6 } S { ∈
(A,b) : rk(A ) = rk(A) , so that (A) = (A,0), (A) = (A,0) and (A) = (A,0). We
p(x) 0 0 1 1
S } S S S S S S
remark that by elementary properties of systems of linear equations, we have the trivial upper bound
(A,b) [n]m (A,b) [n]m (A,b) [n]m nm−rk(A). (9)
0 1
S ∩ ≤ S ∩ ≤ S ∩ ≤
The next lemma(cid:12)is due to Janson(cid:12) an(cid:12)d Rucin´ski [20](cid:12)and(cid:12)establishes a lo(cid:12)wer bound that matches this
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
up to a constant for homogeneous systems. It could be trivially extended to include non-homogeneous
systems.
Lemma 2.3 (Janson and Rucin´ski [20]). Let r,m N and a matrix A (Z) be given. If
r×m
∈ ∈ M
(A) Nm is non-empty then there exists a constant c =c (A)>0 such that
0 0 0
S ∩
(A) [n]m (A) [n]m (A) [n]m c nm−rk(A). (10)
1 0 0
|S ∩ |≥|S ∩ |≥|S ∩ |≥
4
Inordertodeterminetheexactasymptoticvalueof (A) [n]m /nm−rk(A)or (A) [n]m /nm−rk(A),
0
|S ∩ | |S ∩ |
one needs to employ Ehrhart’s Theory, see for example Ru´e et al. [14].
Lastly, let P(A) = p(x) : x (A) denote the family of all set partitions of the column indices
1
{ ∈ S }
[m] stemming from non-trivial solutions. The following lemma gives us the necessary tool to handle
non-trivial solutions with repeated entries.
Lemma 2.4. For every r,m N, A (Z), partition p P(A) and set T N we have
r×m
∈ ∈M ∈ ⊂
x (A) Tm :p(x)=p (A ) T|p| . (11)
1 0 p
{ ∈S ∩ } ≤ S ∩
(cid:12) (cid:12) (cid:12) (cid:12)
Proof. Write p = T1,...,T(cid:12)s for some 1 s m su(cid:12)ch t(cid:12)hat min(T1) (cid:12)< < min(Ts). Let Q =
{ } ≤ ≤ ···
min(T ),...,min(T ) . Now for every x = (x ,...,x ) (A) Tm such that p(x) = p, we would
1 s 1 m 1
{ } ∈ S ∩
have xQ =(x ,...,x ) T|p| as well as xQ (A ) as can be readily seen by the definition
min(T1) min(TS) ∈ ∈S p
of A . Since p = p(x), the vector xQ would furthermore be proper, so that xQ (A ) T|p|. The
p 0 p
∈ S ∩
map x (A) : p(x) = p Tm (A ) T|p|, x xQ is clearly injective, proving the desired
1 0 p
{ ∈ S }∩ → S ∩ 7→
statement.
An easy corollary of this result is clearly that if there are no proper solutions to the system A in a
p
set, then there can also not be non-trivial solutions to A whose repetitions are indicated by p.
2.2 Subsystems The notion of subsystems was originally introduced by Ro¨dl and Rucin´ski [6] when
developing a sparse version of Rado’s Partition Theorem. Recall the definitions from the introduction,
especially r =rk(A) rk(AQ). Observe that we can without loss of generality assume that A is of full
Q
−
rank for this part, since the solution space is unaffected by this assumption. This will simplify notation
significantly.
For a given matrix A (Z) and column indices Q [m], we will now construct through
r×m
∈ M ∅ ⊆ ⊆
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 ,a ,...,a so that the rows of AQ and AQ are
1 2 r
respectively aQ,aQ,...,aQ and aQ,aQ,...,aQ. Here we allow for empty vectors and matrices. If
1 2 r 1 2 r
rk(AQ) < rk(A), then we can express exactly r > 0 of the r rows of AQ as linear combinations of
Q
the rest, that is there are indices i < < i [m] and integers d ,dj Z for i i ,...,i and
1 ··· rQ ∈ i i ∈ ∈ { 1 rQ}
j [m] i ,...,i so that
∈ \{ 1 rQ}
d aQ = djaQ for i i ,...,i . (12)
i i i j ∈{ 1 rQ}
j∈[m]\X{i1,...,irQ}
Consider now the following integer-valued matrix with r rows and Q columns
Q
| |
d aQ dj aQ
i1 i1 − i1 j
 j∈[m]\.{Pi1,...,irQ} 
A[Q]= .. ∈MrQ×|Q|(Z). (13)
 d aQ dj aQ 
 irQ irQ −j∈[m]\{Pi1,...,irQ}irQ j 
 
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
———— rk(A) r
Q
−
B = A[Q] 0 r (14)
  (cid:3) Q (cid:12)
0 0 r rk(A) (cid:12)
(cid:12) (cid:3) −(cid:12)
 (cid:12)  (cid:12)
through elementary row operations, that is A = P(cid:12)−1 B(cid:3) where P (cid:12) is an invertible rectangular
(cid:12)1 · 1(cid:12)∈ Mr×r
matrix. We have (B)= (A), that is the homogeneous solution space remains unchanged, at least up
S S
to the column permutation necessary to ensure that Q= 1,..., Q .
{ | |}
5
ObservethatthematrixA[Q]isonlywelldefineduptoourchoicesofindicesi andcoefficientsd ,dj.
k i i
However,thehomogeneoussolutionspace (A[Q]) isindependentofthese,sowepickonerepresentative
S
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).
Remark 2.5. A[Q] is of full rank, that is rk(A[Q]) = r for any Q [m] satisfying r > 0. If A was
Q Q
⊆
irredundant, positive or abundant, then A[Q] trivially also fulfils these properties for any Q [m] such
⊆
that r >0.
Q
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.
Lemma 2.6 (Kusch et al. [21]). For any r,m N, abundant matrix A (Z) and selection of
r×m
∈ ∈ M
column indices Q [m] the following holds. If Q 2 then we have Q r 1 > 0, that is the
Q
⊆ | | ≥ | |− −
parameter m (A) is well-defined. If Q =1 then we have r =0.
1 Q
| |
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 canbe found in Kuschet al.[21]. Note that this was previously provenby Ro¨dl and Rucin´ski
for proper solutions [6].
Lemma 2.7 (Kusch et al. [21]). For any r,m N, matrix A (Z) and set T N the following
r×m
∈ ∈M ⊂
holds. If there exists a selection of column indices Q [m] such that r > 0 and (A[Q]) T|Q| =
Q 1
⊆ S ∩ ∅
then (A) Tm = .
1
S ∩ ∅
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.
Proposition 2.8. For every ǫ > 0, r,m N such that m 2 and matrix A (Z) the following
r×m
∈ ≥ ∈ M
holds. If A is irredundant, positive but not abundant, then we have lim P([n] ⋆ A) = 0 for any
n→∞ p →ǫ
p(n)=o(1).
Proof. SinceAisnotabundantbutpositiveandirredundant,thereexistssomeQ [m]satisfying Q =2
⊆ | |
such that A[Q] = ( a b ) for some a,b N, a = b. By Lemma 2.7 we can replace A with A[Q]. It
− ∈ 6
followsbyEquation(9),Lemma2.3aswellasthelinearityofexpectationthatE (A) [n]2 =Θ(np2)
|S ∩ p|
whileE([n] )=np. Ifnp=O(1)thenE (A) [n]2 =o(1)andtheresulttrivallyholdsbyMarkov’s
| p| |S ∩ p| (cid:0) (cid:1)
Inequality. If np then by Chernoff [n] np/2 asymptotically almost surely. Since p = o(1) we
→ ∞ (cid:0)| p| ≥ (cid:1)
have E (A) [n]2 = o(np/2) and therefore for any given set of positive density, we can remove one
|S ∩ p|
element per solutionand still asymptotically almost surely have a solution-free set of that same density.
(cid:0) (cid:1)
This proves the desired result.
2.3 RemovalLemmaandSupersaturationResults Acommoningredienttoprovingsparseresults
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 Szemer´edi’s Theorem which states that a set of positive density contains not just one,
butapositiveproportionofallk–termarithmeticprogressions. Frankl,GrahamandRo¨dl[4]formulated
such results for both for partition and density regular systems.
Lemma 2.9 (Theorem 1 in Frankl, Graham and Ro¨dl [4]). For a given partition regular matrix A
∈
(Z) and s N there exists ζ = ζ(A,s) > 0 such that for any partition [n] = T ˙ ... ˙ T and n
r×m 1 s
M ∈ ∪ ∪
large enough we have (A) Tm + + (A) Tm ζ (A) [n]m .
|S0 ∩ 1 | ··· |S0 ∩ s |≥ |S0 ∩ |
6
Lemma 2.10 (Theorem 2 in Frankl, Graham and Ro¨dl [4]). For a given density regular matrix A
∈
(Z) and δ > 0 there exists ζ = ζ(A,δ) > 0 such that any subset T [n] satisfying T δn
r×m
M ⊆ | | ≥
contains at least ζ (A) [n]m proper solutions for n large enough.
0
|S ∩ |
We will extend Lemma 2.10 to cover the scope of this note by using an arithmetic removal lemma.
Green[23] first formulatedsucha statement for linear equations in anabelian group. Later Shapira[24]
as well as independently Kr´al’, 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 be the finite field of order q. Let X F be a subset of
q q
⊂
F and A (F ) a matrix of full rank. For = x Fm : A xT = 0T and every ǫ > 0 there
q ∈ Mr×m q S { ∈ q · }
exists an η =η(ǫ,r,m) such that if Xm <η then there exists a set X′ X with X′ <ǫq and
|S∩ | |S| ⊂ | |
(X X′)m = .
S∩ \ ∅
Applying this result, we formulate the following extension of Lemma 2.10.
Lemma 2.12 (Supersaturation). For a given r,m N, positive and irredundant matrix A (Z)
r×m
∈ ∈M
and δ > π(A) there exists ζ = ζ(δ,A) > 0 such that any subset T [n] satisfying T δn contains at
⊆ | | ≥
least ζ (A) [n]m proper solutions for n large enough.
0
|S ∩ |
Proof. Let q = q(A,n) be a prime number between 2mnmax(A) and 4mnmax(A) and F the finite
q
| | | |
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 =Z and we can
q ∼ q
identify the integers with their corresponding residue classes in F . The matrix A now defines a map
q
from Fm to Fr. A solution in (A) clearly lies in the and, as we have chosen q large enough, all
q q S S
canonical representatives from [n]m also lie in (A) [n]m for n max A.
S∩ S ∩ ≥ | |
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
⊆ | | ≥
considerthecorrespondingsetX ofresidueclassesinF . Oneneedstoremoveatleast(δ δ′)nelements
q
−
from T in order for Tm to avoid (A) in [n], so one needs to remove at least an
0
S
(δ δ′)n (δ δ′)
ǫ= − − >0
q ≥ 4mmax(A)
| |
proportion of elements in F from X so that Xm avoids in F . It follows from Theorem 2.11 that
q q
S
Xm η for some η =η(ǫ,r,m). Since we havechosen q largeenough, it follows that T contains
|S∩ |≥ |S|
at least an η proportion of (A) [n]m. An easy consequence of Equation (9) and Lemma 2.3 is that
S ∩
lim (A) [n]m / (A) [n]m c for c =c (A)> 0 as given by Lemma 2.3. It follows result
n→∞ 0 0 0 0
|S ∩ | |S ∩ |≥
holds for n large enough and ζ =ζ(δ,A)=(c η)/2.
0
2.4 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 we denote its vertex set by V( ) and its set of hyperedges by E( ). The
H H H
cardinalityofthesesetswillberespectivelydenotedbyv( )ande( ). GivensomesubsetofverticesA
H H ⊆
V( )wedenotethesubgraphitinducesin by [A]anditsdegreebydeg (A)= e E( ):A e .
H H H H |{ ∈ H ⊆ }|
For ℓ N we denote the maximum ℓ-degree by ∆ ( ) = max deg (A) : A V( ) and A = ℓ . Let
∈ ℓ H { H ⊆ H | | }
the set of independent vertex sets in be denoted by ( ). Lastly, let be a uniform hypergraph,
H I H H F
an increasing family of subsets of V( ) and ǫ > 0. We say that is ( ,ǫ)-dense if e( [A]) ǫe( )
H H F H ≥ H
for every A .
∈F
Theorem 2.13 (Hypergraph Containers, Theorem 2.2 in [9]). For every m N, c>0 and ǫ>0, there
∈
exists a constant C = C(m,c,ǫ) > 0 such that the following holds. Let be an m-uniform hypergraph
H
7
and let 2V(H) be an increasing family of sets such that A ǫv( ) for all A . Suppose that
F ⊆ | |≥ H ∈F H
is ( ,ǫ)-dense and p (0,1) is such that, for every ℓ 1,...,k ,
F ∈ ∈{ }
e( )
∆ ( ) cpℓ−1 H . (15)
ℓ
H ≤ v( )
H
Then there exists a family V(H) and functions f : and g : ( ) such that for
T ⊆ ≤Cpv(H) T → F I H → T
every I ( ),
∈I H (cid:0) (cid:1)
g(I) I and I g(I) f(g(I)). (16)
⊆ \ ⊆
Thestatementgivestheexistenceofasmallnumberofcontainers andsomefingerprints sothat
F T
every independent set I in is identified with a fingerprint g(I) that determines a container f(g(I))
H
which contains the independent set.
Next, let = ( ) be a sequence of m-uniform hypergraphs and let α [0,1). We say that
n n∈N
H H ∈
is α-dense if for every δ > 0, there exist some ǫ > 0 such that for U V( ) which satisfies
n
H ⊆ H
U >(α+δ)v( )we havee( [U])>ǫe( ) forn largeenough. Balogh,MorrisandSamotijproved
n n n
| | H H H
the following consequence of their container statement.
Theorem 2.14 (Sparse Sets through Hypergraph Containers, Theorem 5.2 in [9]). Let = ( )
n n∈N
H H
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
∈{ }
e( )
∆ ( ) Cq(n)ℓ−1 Hn . (17)
ℓ n
H ≤ v( )
n
H
If is α-dense, then for everyδ >0, thereexists aconstantc=c(C,α,m)>0suchthat if p(n)>cq(n)
H
and p(n)v( ) as n , then asymptotically almost surely
n
H →∞ →∞
α [V( ) ] (α+δ)p(n)v( ). (18)
n n p(n) n
H H ≤ H
(cid:0) (cid:1)
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.
3 Proof of the 1-statement in Theorem 1.2
Let be the hypergraph with vertex set V( )=[n] and edge multiset
n n
H H
E( )= x ,...,x :(x ,...,x ) (A) [n]m .
n 1 m 1 m 0
H { } ∈S ∩
(cid:8)(cid:8) (cid:9)(cid:9)
Observe that H can be a multigraph, that is multiple edges are allowed, but the multiplicity of each
n
edge is clearly bounded by m!. We do this to simplify counting, since this way we have E( ) =
n
| H |
(A) [n]m . Weobservethatwecanlimitourselvestopropersolutionswhenprovingthe1-statement.
0
|S ∩ |
Corollary 2.12 now states that = ( ) is π(A)-dense. In order to apply Theorem 2.14, it
n n∈N
H H
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 .
n
H
Lemma 3.1. For 1 ℓ m we have ∆ℓ( n) ℓ!mℓmaxQ⊆[m],|Q|=ℓn(m−rk(A))−(|Q|−rQ).
≤ ≤ H ≤
Proof. For =( ) as defined above and ℓ 1,...,m we have
n
H H ∈{ }
∆ ( ) max x (A) [n]m : Q [m],π S(ℓ) s.t. xQ =(x ,...,x )
ℓ n 0 π(1) π(ℓ)
H ≤x1,...,xℓ∈[n] { ∈S ∩ ∃ ⊆ ∈ }
m (cid:12) (cid:12)
ℓ! (cid:12)max x [n]m−l AQ xT = AQ (x ,...,x )T (cid:12)
1 ℓ
≤ (cid:18)ℓ(cid:19)(x1,...,xℓ)∈[n]ℓ { ∈ | · − · }
Q⊆[m],|Q|=ℓ (cid:12) (cid:12)
(cid:12) (cid:12)
8
ℓ!mℓ max max (AQ,b) [n]m ℓ!mℓ max n|Q|−rk(AQ)
≤ Q⊆[m] b∈Zr S ∩ ≤ Q⊆[m]
|Q|=ℓ (cid:12) (cid:12) |Q|=l
(cid:12) (cid:12)
=ℓ!mℓ max n(m−rk(A))−(|Q|−rQ)
Q⊆[m]
|Q|=l
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 = 0 for any Q [m] satisfying Q = 1 due to Lemma 2.6 and that there exists
Q
⊆ | |
c = c (A) > 0 such that e( ) c nm−rk(A) due to Lemma 2.3. Using Lemma 3.1 we now observe
0 0 n 0
H ≥
that
e( )
∆ ( ) mnm−rk(A)−1 m/c Hn .
1 n 0
H ≤ ≤ v( )
n
H
For ℓ 2,...,m we again apply Lemma 3.1 to see that
∈{ }
∆ℓ( n) ℓ!mℓ max n(m−rk(A))−(|Q|−rQ) =ℓ!mℓ max n−|Q||Q−|r−Q1−1 ℓ−1 nm−rk(A)−1
H ≤ Q⊆[m],|Q|=ℓ Q⊆[m],|Q|=ℓ
(cid:16) (cid:17)
ℓ!mℓ n−1/m1(A) ℓ−1 nm−rk(A)−1 (ℓ!mℓ)/c n−1/m1(A) ℓ−1 e(Hn).
0
≤ ≤ v( )
n
H
(cid:0) (cid:1) (cid:0) (cid:1)
Lastly we observe that n−1/m1(A)v( n)=n1−1/m1(A) as m1(A)>1. It follows that the prerequi-
H →∞
sitesofTheorem2.14holdforC =(m!mm)/c0, q =q(n)=n−1/m1(A) andwecanchoosethe 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.
4 A Short Proof of the 1-statement in Theorem 1.1
Asstatedintheintroduction,thisresultwaspreviouslyprovenbyFriedgut,Ro¨dlandSchacht[7]aswell
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.
Corollary 4.1. For a given partition regular matrix A (Z) and s N there exist ǫ = ǫ(A,s)
r×m
∈ M ∈
and δ = δ(A,s) > 0 such that for any T ,...,T [n] satisfying (A) Tm ǫ (A) [n]m for
1 s ⊆ |S0 ∩ i | ≤ |S0 ∩ |
1 i s we have [n] (T T ) δn for n large enough.
1 s
≤ ≤ \ ∪···∪ ≥
Proof. Let ζ = ζ((cid:12)A,s + 1) be as in(cid:12)Lemma 2.9 and ǫ = ǫ(A,s) = ζ/2s. Set T˜ = T i−1T for
(cid:12) (cid:12) i i\ j=1 j
1 i s and T˜ =[n] r T and consider the partition [n]=T˜ ˙ ... ˙ T˜ ˙ T˜ . By Lemma 2.9
≤ ≤ s+1 \ j=1 j 1∪ ∪ s∪ s+1 S
we have (A) T˜m + + (A) T˜m ζ (A) [n]m and since by assumption (A) T˜m
|S0 ∩ 1 | ···S |S0 ∩ r+1|≥ |S0 ∩ | |S0 ∩ i |≤
(A) Tm ζ/2s (A) [n]m for all i 1,...,s , we have (A) ([n] (T T ))m
|S0 ∩ i | ≤ |S0 ∩ | ∈ { } |S0 ∩ \ 1∪···∪ s | ≥
ζ/2 (A) [n]m . ObservethatbyLemma3.1everyelementin[n]iscontainedinatmostmnm−rk(A)−1
0
|S ∩ |
solutions and by Lemma 2.3 there exists c = c (A) > 0 such that (A) [n]m c nm−rk(A) for n
0 0 0 0
|S ∩ | ≥
large enough, so that the results follows for δ =ζc /2.
0
ThesecondingredientisstatedinthefollowingcorollarythatisobtainedbyapplyingtheHypergraph
Container Theorem to the hyperpgraph of solutions.
Corollary 4.2. For a given partition regular matrix A (Z) and ǫ > 0 there exist t = t(n) sets
r×m
∈ M
T ,...,T [n] for some c >0 as well as sets C ,...,C [n] such that
1 t ∈ ≤c0n1−1/m1(A) 0 1 t ⊆
(cid:0) (cid:1) (A) Cm ǫ (A) [n]m . (19)
|S0 ∩ i |≤ |S0 ∩ |
9
Furthermore, for every set T [n] satisfying (A) Tm = there exists 1 i t such that
0
⊆ S ∩ ∅ ≤ ≤
T T C . (20)
i i
⊆ ⊆
Proof. Let again be the hypergraph with vertex set V( )=[n] and edge multiset
n n
H H
E( )= x ,...,x :(x ,...,x ) (A) [n]m .
n 1 m 1 m 0
H { } ∈S ∩
We have previously observed th(cid:8)a(cid:8)t there exists a c > 0 such that ∆ c(cid:9)p(cid:9)(n)ℓ−1e( )/v( ) for
ℓ n n
≤ H H
p = p(n)= n−1/m1(A). We observe that n is trivially ( ,ǫ)-dense for = T [n]: 0(A) Tm
H F F { ⊆ |S ∩ |≥
ǫ (A) [n]m . Applying Theorem 2.13 gives the desired statement.
0
|S ∩ |}
We are now ready to give a short proof of the 1-statement in Theorem 1.1 following the ideas of
NenadovandSteger[8]. Letǫ,δ >0be asinCorollary4.1andlett=t(n), c , S ,...,S andC ,...,C
0 1 t 1 t
be as in Corollary 4.2. Let C =C(A,s) 2sc /δ be constant.
0
≥
Observenow that for a partition of the randomset T ˙ ... ˙ T =[n] satisfying (A) Tm = for
1∪ ∪ s p S0 ∩ i ∅
all i 1,...,s there exist j ,...,j 1,...,t so that S T C for all i 1,...,s . Since
∈ { } 1 s ∈ { } ji ⊆ i ⊆ ji ∈ { }
T [n] for 1 i s and [n] (C C ) [n] = we can bound the probability of [n] not
i p 1 s p p
⊆ ≤ ≤ \ ∪···∪ ∩ ∅
fulfilling the partition property by
P([n] A) P(S ,...,S [n] [n] (C ,...,C ) [n] = ).
p 6→s ≤ j1 js ⊆ p ∧ \ j1 js ∩ p ∅
j1,...,jXs∈{1,...,t}
ObservethatthetwoeventsS ,...,S [n] and[n] (C ,...,C ) [n] = areindependent,sothat
j1 js ⊆ p \ j1 js ∩ p ∅
we have
P([n]p s A) p|Ssj=1Sj|(1 p)|[n]\(Cj1,...,Cjs)|.
6→ ≤ −
j1,...,jXs∈{1,...,t}
We bound this by choosing k =| sj=1Sj|≤sc0n1−1/m1(A), then picking k elements and lastly deciding
for each element in this selection in which of the S it is contained, so that we have
S i
sc0n1−1/m1(A) n sc0C−1np e2snp k
P([n] A) (1 p)δn (2s)kpk e−δnp .
p s
6→ ≤ − k ≤ k
k=0 (cid:18) (cid:19) k=0 (cid:18) (cid:19)
X X
Lastly we note that for c>0 the function f(x)=(c/x)x is increasing for 0 x c/e since d/dxf(x)=
≤ ≤
(c/x)x(log(c/x) 1). We have chosen C large enough so that for n large enough we have
−
k
e2sc0C−1np
P([n] A) e−δnp(sc C−1np+1) e−δnp eδnp/2 =o(1).
p 6→s ≤ 0 sc0C−1 ! ≤
As desired it follows that [n]p s A asymptotically almost surely for p Cn−1/m1(A).
→ ≥
5 Proof of Theorem 1.3 – A Rado-type 0-statement
LetQ [m]beasetofcolumnindicessatisfying Q 2suchthat(Q 1)/(Q r 1)=m (A). Due
Q 1
⊆ | |≥ | |− | |− −
toLemma2.7wecanreplaceAwithA[Q]ifnecessaryinordertoguaranteethat(m 1)/(m rk(A) 1)=
− − −
m (A). Due to Lemma 2.4 we know that
1
P([n] ⋆ A) P [n] A P [n] A . (21)
p →s ≤  p →s p ≤ p →s p
p∈[P(A)(cid:16) (cid:17) p∈XP(A) (cid:0) (cid:1)
 
Let us bound the individual probabilities P [n] A for each p P(A). For p = m, that is
p s p
→ ∈ | |
p = 1 ,..., m , we know due to Ro¨dl and Rucin´ski’s Theorem 1.1 that there exists a c = c(A,s)
{{ } { }} (cid:0) (cid:1)
10