The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs

The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs tài liệu, giáo án, bài...

The sovability of norm, bilinear and quadratic

equations over finite fields via spectra of graphs

Le Anh Vinh

Communicated by Christopher D Sogge

Abstract In this paper we will give a unified proof of several results on the sovability ofsystems of certain equations over finite fields, which were recently obtained by Fourieranalytic methods Roughly speaking, we show that almost all systems of norm, bilinear orquadratic equations over finite fields are solvable in any large subset of vector spaces overfinite fields

Keywords Bilinear equations, quadratic equations, spectral graphs, finite fields

2010 Mathematics Subject Classification 11L40, 11T30

1 Introduction

The main purpose of this paper is to give a unified proof of several results on thesolvability of systems of certain equations over finite fields, which were recentlyobtained by Fourier analytic methods We will see that after appropriate graph the-oretic results are developed, many old and new results immediately follows In thissection, we discuss the motivation and background results for our work

Let Fqdenote a finite field with q elements, where q, a power of an odd prime, isviewed as an asymptotic parameter ForE Fqd (d  2), the finite analogue of theclassical Erd˝os distance problem is to determine the smallest possible cardinality

of the set

.E/D ¹kx yk D x1 y1/2C


C xd yd/2W x; y 2 Eº  Fq:The first non-trivial result on the Erd˝os distance problem in vector spaces over fi-nite fields is due to Bourgain, Katz, and Tao ([9]), who showed that if q is a prime,

q  3 (mod 4), then for every " > 0 and E  Fq2withjEj  C"q2, there exists

ı > 0 such that j.E/j  Cıq1Cı for some constants C"; Cı The relationshipbetween " and ı in their arguments, however, is difficult to determine In addition,

it is quite subtle to go up to higher dimensional cases with these arguments vich and Rudnev ([19]) used Fourier analytic methods to show that there are abso-lute constants c1; c2> 0 such that for any odd prime power q and any set E  Fqd

Iose-of cardinalityjEj  c1qd=2, we have

Iosevich and Rudnev reformulated the question in analogy with the Falconer tance problem: how large does E  Fqd, d  2, needed to be ensure that .E/contains a positive proportion of the elements of Fq The above result implies that

dis-ifjEj  2qdC12 , then .E/D Fqdirectly in line with Falconer’s result in ean setting that for a setE with Hausdorff dimension greater than d C 1/=2 thedistance set is of positive measure At first, it seemed reasonable that the exponent.d C 1/=2 may be improvable, in line with the Falconer distance conjecture de-scribed above However, Hart, Iosevich, Koh and Rudnev discovered in [12] thatthe arithmetic of the problem makes the exponent d C 1/=2 best possible in odddimensions, at least in general fields In even dimensions it is still possible thatthe correct exponent is d=2, in analogy with the Euclidean case In [10], Chapman

Euclid-et al took a first step in this direction by showing that if a sEuclid-etE  Fq2satisfiesjEj  q4=3, thenj.E/j  cq This is in line with Wolff’s result for the Falconerconjecture in the plane which says that the Lebesgue measure of the set of dis-tances determined by a subset of the plane of Hausdorff dimension greater than4=3 is positive

In [30], the author gave another proof of (1.1) using the graph theoretic method(see also [37] for a similar proof) The (common) main step of these proofs is to es-timate the number of occurrences of a fixed distance It was shown that for a fixeddistance, given that the point set is large, the number of occurrences of any fixeddistance is close to the expected number This implies that there are many distinctdistances occur in a large point set In the case of real number field, most of theknown results, however, are actually proved in a stronger form In order to showthat there are at least g.n/ distinct distances determined by an n-point set in theplane, one usually proves that for any n-point set P , there exists a point p 2 P thatdetermines at least g.n/ distinct distances to P Chapman et al ([10]) obtained ananalogous result in the finite field setting They also proved a similar result for thepinned dot product sets …y.E/ D ¹x
y W x 2 Eº In this paper, we will derivethese results using spectral graph methods

A classical result due to Furstenberg, Katznelson and Weiss ([13]) states that if

E  R2of positive upper Lebesgue density, then for any ı > 0, the hood ofE contains a congruent copy of a sufficiently large dilate of every three-point configuration An example of Bourgain ([7]) showed that it is not possible toreplace the thickened setEı byE for arbitrary three-point configurations In thecase of k-simplex, that is the kC 1 points spanning a k-dimensional subspace,Bourgain ([7]), using Fourier analytic techniques, showed that a setE of positive

ı-neighbor-upper Lebesgue density always contains a sufficiently large dilate of every degenerate k-point configuration where k < d In the case k D d , the problemstill remains open Using Fourier analytic method, Akos Magyar ([23, 24]) con-sidered this problem over the integer lattice Zd He showed that a set of positivedensity will contain a congruent copy of every large dilate of a non-degeneratek-simplex where d > 2kC 4.

non-Hart and Iosevich ([18]) made the first investigation in an analog of this question

in finite field geometries Let Pkdenote a k-simplex Given another k-simplex Pk0,

we say Pk  Pk0 if there exist  2 Fqd, and O 2 SOd.Fq/, the set of d -by-d thogonal matrices over Fq, such that Pk0 D O.Pk/C  Under this equivalentrelation, Hart and Iosevich ([18]) observed that one may specify a simplex by thedistances determined by its vertices They showed that ifE  Fqd (d  kC12
) is

or-a set of cor-ardinor-alityjEj & C qkC1kd C k

, thenE contains a congruent copy of everyk-simplices (with the exception of simplices with zero distances) Using graph the-oretic method, the author ([35]) showed that the same result holds for d  2k andjEj  qd 1Ck Here, and throughout, X Y means that there exists C > 0 suchthat X  C Y , and X  Y means that X D o.Y / Note that serious difficultiesarise when the size of simplex is sufficiently large with respect to the ambient di-mension Even in the case of triangles, the result in [35] is only non-trivial for

d  4 Covert, Hart, Iosevich, and Uriarte-Tuero ([11]) addressed the case of angles in plane over finite fields They showed that ifE has density  for some

tri-C q 1=2    1 with a sufficiently large constant C > 0, then the set of trianglesdetermined byE, up to congruence, has density c In [36], the author studiedthe remaining case: triangles in three-dimensional vector spaces over finite fields.Using a combination of graph theory method and Fourier analytic techniques, theauthor showed that ifE Fqd (d  3) of cardinality jEj & C qdC22 , the set of tri-angles, up to congruence, has density greater than c Using Fourier analytic tech-niques, Chapman et al ([10]) extended this result to higher dimensional cases.More precisely, they showed that ifjEj & qdCk2 (d  k), then the set of k-simp-lices, up to congruence, has density greater than c They also obtained a strongerresult whenE is a subset of the d -dimensional unit sphere

Sd D ¹x 2 Fqd W kxk D 1º:

In particular, it was proven ([10, Theorem 2.15]) that if E  Sd of cardinalityjEj & qdCk 12 , then E contains a congruent copy of a positive proportion of allk-simplices In this paper, we will obtain similar results in a more general setting.Let Q be a non-degenerate quadratic form on Fqd The Q-distance between twopoints x; y 2 Fqd is defined by Q.x y) We consider the systemL of k2
equa-tions

Q.xi xj/D ij; xi 2 E; i D 1; : : : ; k; (1.2)

over Fqd, with variables from arbitrary setE Fqd We show that

 ifjEj  qd 1Ck 1, then the system (1.2) is solvable for all ij 2 F

q,

 ifjEj  q.d Ck/=2, then that system 1.2 is solvable for at least 1 o.1//q.k/possible choices of ij 2 Fq

A related question that has recently received attention is the following Letting

A Fq, how large doesA need to be to ensure that Fq A
A C


C A
A (dtimes) Bourgain ([8]) showed that ifA  Fqof cardinalityjAj  C q3=4, then

A
A C A
A C A
A D Fq Glibichuk and Konyagin ([16]) proved in the case

of prime fields Zp that for d D 8, one can take jAj >pq Glibichuk ([15]) thenextended this result to arbitrary finite fields Note that this question can be stated

in a more general setting LettingE Fqd, how large doesE need to be to ensurethat the equation

x
y D ; x; y 2 E;

is solvable for any given 2 Fq Hart and Iosevich ([18]), using exponential sums,showed that one can takejEj > q.d C1/=2for any d  2 In this paper, we will giveanother proof of this result using spectral graph methods

In analogy with the study of simplices in vector spaces over finite fields, the thor ([31]) studied the sovability of systems of bilinear equations over finite fields.More precisely, for any non-degenerate bilinear form B.
;
/ in Fqd, we considerthe following system of l  k2
equations:

au-B.ai; aj/D ij; ai 2 E; i D 1; : : : ; k; (1.3)over Fqd, with variables from an arbitrary setE  Fqd Using character sum ma-chinery and methods from graph theory, the author ([31]) showed that if each vari-able in the system (1.3) appears in at most t  k 1 equations and jEj  qd 1Ct,then for any ij 2 Fq, the system (1.3) has 1C o.1//q ljEjk solutions Again,serious difficulties arise when the number of equations that each variable involves

is sufficiently large with respect to the ambient dimension In particular, that result

is only non-trivial in the range of d  2t In the case of three variables and threeequations, the author also proved ([31, Theorem 1.4]) that the system (1.3) is solv-able for 1 o.1//q3triples 12; 23; 31/2 Fq/3ifjEj  qdC22 In this paper,

we will extend this result to systems with many variables More precisely, we willshow that if E Fqd is a set of cardinality jEj  qdCk2 , then the system (1.3)

of all k2
equations is solvable for 1 o.1//q.k2/ possible choices of ij 2 Fq,

1 i < j  k

We remark here that one can also obtain this result using Fourier analytic ods (for example, using [10, Theorem 2.14] instead of [10, Theorem 2.12] in the

meth-proof of [10, Theorem 2.13]) However, techniques involved in difference lems are considerable in Fourier analytic proofs The main advantage of our ap-proach is that we can obtain all the aforementioned results at once, after comput-ing the eigenvalues of appropriate graphs We will also demonstrate our method

prob-by some related results on norm equations and sum-product equations over finitefields

Theorem 2.1 ([21, Theorem 4.10]) Let H be a fixed graph with r edges, s tices, and maximum degree, and let G D V; E/ be an n; d; /-graph where

ver-d  0:9n Let m < n satisfy m  .n=d / Then, for every subsetV0  V ofcardinalitym, the number of (not necessarily induced) copies of H in V0is

s

j Aut.H /j

 dn

r

:

If we are only interested in the existence of one copy of H , then one can times improve the conditions on d and  in Theorem 2.1 The first result of thispaper is an improvement of the conditions on d and  in Theorem 2.1 for com-plete bipartite graphs Let G G be the bipartite graph with two identical vertexparts V G/ and V G/ Two vertices u and v in two different parts are connected

some-by an edge if and only if they are connected some-by an edge in G For any two subsets

U1; U2  V G/, let GŒU1; U2 be the induced bipartite subgraph of G  G on

U1 U2

Theorem 2.2 For anyt  s and t  2, let G D V; E/ be an n; d; /-graph.For every subsetsU1; U2 V with

jU1jjU2j  2.n=d /t Cs;

the induced subgraphGŒU1; U2 contains

.1C o.1//jU1j

sjU2jtsŠt Š

 dn

st

copies ofKs;t

Note that the bound in Theorem 2.2 is stronger than that in Theorem 2.1 when

t > s For small bipartite subgraphs, K2;t, we can further improve the bound inTheorem 2.2

Theorem 2.3 For anyt  1, let G D V; E/ be an n; d; /-graph For every setsU1; U2 V with

sub-jU1jjU2j  2.n=d /t C1the induced subgraphGŒU1; U2 contains

.1C o.1//jU1j

sjU2jt2Št Š

 dn

Theorem 2.4 LetH be a fixed edge-colored graph with r edges, s vertices, andmaximum degree , and let GD V; E/ be an n; d; /-colored graph, where

d  0:9n Let m < n satisfy m  .n=d / For every subsetV0  V of nalitym, the number of (not necessarily induced) copies of H in V0is

s

j Aut.H /j

 dn

r

:

Theorem 2.5 For any t  2, let H be a fixed edge-colored complete bipartitegraphKs;t withs  t Let G D V; E/ be an n; d; /-colored graph For everysubsetsU1; U2 V with

jU1jjU2j  2.n=d /t Cs;the induced subgraphGŒU1; U2 contains

.1C o.1//jU1j

sjU2jtAut.H /

 dn

st

copies ofH

Theorem 2.6 For any t  1, let H be a fixed edge-colored complete bipartitegraphK2;t, and letGD V; E/ be an n; d; /-colored graph For every subsets

U1; U2 V with

jU1jjU2j  2.n=d /t C1;the induced subgraphGŒU1; U2 contains

.1C o.1//jU1j

2jU2jtAut.H /

 dn

2t

copies ofH

The proof of Theorem 2.4 is similar to that of [21, Theorem 4.10], the proofs ofTheorem 2.5 and Theorem 2.6 are similar to the proofs of Theorem 2.2 and Theo-rem 2.3, respectively To simplify the notation, we will only present the proofs

of single-color results Note that going from single-color formulations rems 2.1, 2.2 and 2.3) to multi-color formulations (Theorems 2.4, 2.5 and 2.6) isjust a matter of inserting different letters in a couple of places

(Theo-Although we cannot improve the conditions on d and  in Theorem 2.4 (orequivalently Theorem 2.1), we will show that if the number of colors is large,under a weaker condition, any large induced subgraph of an n; d; /-color graphscontains almost all possible colorings of small complete subgraphs

Theorem 2.7 For anyt  2, let G D V; E/ be an n; d; /-colored graph, andletm < n satisfy m .n=d /t =2 Suppose that the color setC has cardinalityjCj D 1 o.1//n=d Then for every subset U  V with cardinality m, the in-duced subgraph ofG on U contains at least 1 o.1//jCj.2t/ possible colorings

ofKt

The results above could also be considered as a contribution to the ing comprehensive study of graph theoretical properties of n; d; /-graphs, whichhas recently attracted lots of attention both in combinatorics and theoretical com-puter science For a recent survey about these fascinating graphs and their proper-ties, we refer the interested reader to the paper of Krivelevich and Sudakov ([21]).2.2 Norms in sum sets, pinned norms, and norm equations

fast-develop-Let Fqbe a finite field with qD pdelements We denote by NF an algebraic closure

of Fq, and by Fq n  NF the unique extension of the degree n of F for n  1 Theextension Fqn=Fqis a Galois extension, with Galois group Gncanonically isomor-phic to Z=Zn, the isomorphism being the map Z=Zn ! Gn defined by 17! ,

where  is the Frobenius automorphism of Fq ngiven by  X /D Xq Associated

to the extension Fq n=Fq, the norm map N D NFqn=F q W Fqn ! Fqn is defined by

Be-Theorem 2.8 Let2 FqandA; B  Fq n,n 2 Suppose that jAjjBj  qnC2.Then the equationN.XC Y / D  is solvable in X 2 A, Y 2 B

For anyA; B Fq n, define by N.AC B/ the norm set of the sum set, i.e

N.AC B/ D ¹N.X C Y / W X 2 A; Y 2 Bº:

Theorem 2.8 says that ifjAjjBj  qnC2, then Fq  N A C B/ We will showthat under a slightly stronger condition, one can always find many elements X 2 Asuch that the pinned norm set NX.B/, which is defined by

NX.B/D ¹N.X C Y / W Y 2 Bº;

contains almost all elements in Fq

Theorem 2.9 LetA; B  Fq n,n 2 Suppose that A; B satisfy jAj  jBj andjAjjBj  qnC2 Then there exists a subsetA0 ofA with cardinalityjA0j & jAjsuch that for everyX 2 A0, the equationN.XC Y / D  is solvable in Y 2 Bfor at least.1 o.1//q values of 2 F

We also obtain the following results on the solvability of systems of norm tions over finite fields

equa-Theorem 2.10 LetA Fq n,n 2 Consider the systems L of l  2t
norm tions

equa-N.XiC Xj/D ij; Xi 2 A; i D 1; : : : ; t: (2.1)Suppose that each variable appears in at mostk  t 1 equations, and supposethatjAj  qn=2Ct 1 Then for anyij 2 Fq, the above system has

.1C o.1//q ljAjtsolutions

Theorem 2.11 LetA Fq n,n 2 Consider the system (2.1) with 2t
equations.

IfjAj  q.nCt/=2, then that system is solvable for at least.1 o.1//q.2t/ choices

ofij 2 Fq,1 i < j  t

2.3 Dot product set and system of bilinear equations

LetE; F  Fd D Fq


 Fq, d  2 For any non-degenerate bilinear form

B.
;
/ on Fqd, define the product set ofE and F with respect to B by

Note that [18, Theorem 2.1] and [10, Theorem 2.4] are stated only for the dotproduct, but their proofs go through for any non-degenerate bilinear form withoutany essential change As a corollary of our results in Section 2.1, we will givegraph theoretic proofs of Theorems 2.12 and 2.13 In fact, we will prove the fol-lowing result instead of Theorem 2.13

Theorem 2.14 LetE  Fqd,d  2, of cardinality jEj  q.d C1/=2 Then there ists a subsetE0 E of cardinality jE0j D 1 o.1//jEj such that for every y 2 E0,one hasjBy.E/j D 1 o.1//q

ex-Note that the proof of [10, Theorem 2.14] also implies Theorem 2.14 and viceversa We, however, relax the condition onjEj  q.d C1/=2tojEj  q.d C1/=2tosimplify our arguments

In [31], the author studied the solvability of systems of bilinear equations overfinite fields Following the proof of [21, Theorem 4.10], the author proved the fol-lowing result

Theorem 2.15 ([31]) LetE Fqd,d  2 For any non-degenerate bilinear form

B.
;
/ on Fqd, consider the systemsL of l  2t
bilinear equations

B.ai; aj/D ij; ai 2 A; i D 1; : : : ; t: (2.2)Suppose thatjEj  qd 1Ct 1and each variable appears in at mostk  t 1equations Then the system(2.2) is solvable for any ij 2 Fq,1 i < j  t

As a simple consequence of Theorem 2.7 and the construction of product graph

in Section 8, we show that under a weaker condition, sayjAj  q.nCt 1/=2, thesystem (2.2) is solvable for almost all possible choices of parameters ij 2 F.Theorem 2.16 LetE  Fqd,d  2, of cardinality jEj  q.nCt 1/=2 For anynon-degenerate bilinear formB.
;
/ on Fqd, consider the systemsL of t2
bilin-ear equations

B.ai; aj/D ij; ai 2 A; i D 1; : : : ; t: (2.3)Then the above system is solvable for.1 o.1//q.2t/ possible choices of ij 2 F,

Theorem 2.17 ([14]) For anyA Fq, we have

4

q

³:

When one of sum or product sets is small, we have an immediate corollary.Corollary 2.18 Suppose thatA Fqandmin.jA C Aj; jA
Aj/  C jAj for anabsolute constantC > 0.

(i) IfjAj  q2=3, thenmax.jA C Aj; jA
Aj/  q

(ii) IfjAj  q2=3, thenmax.jA C Aj; jA
Aj/  jAj3=q

In [32], the author reproved Theorem 2.17 using standard tools from spectralgraph theory Solymosi gave a similar proof in [29] We will use the same idea tostudy the solvability of sum-bilinear equation

is solvable for any2 Fq

As an easy corollary of Theorem 2.19, we have the following sum-product mate, which can be viewed as an extension of Theorem 2.17

esti-Theorem 2.20 For anyA Fq, let

which implies that

jA
Ajd 1jd Aj  min qjAjd 1;jAj3d 2

Corollary 2.21 LetA be an arbitrary subset of Fqwith cardinalityjAj  q1=2.(i) Suppose thatA satisfiesjA
Aj  C jAj for an absolute constant C > 0 IfjAj  qd=.2d 1/, thenjd Aj  q, and if jAj  qd=.2d 1/, then we have

jd Aj  jAj2d 1=qd 1

(ii) Suppose thatjA C Aj  C jAj for an absolute constant C > 0 and

q2ddC1C1  jAj  q2dd 1:Then we havejA
Aj  jAj.q=jAj/1=d

Using the machinery developed in this paper, we also can study systems ofsum-product equations over finite fields More precisely, we have the followingresult

Theorem 2.22 For anyE Fq Fd and a non-degenerate bilinear formB.
;
/

onFqd, consider the systemL of 2t
equations

2.5 Pinned distances and systems of quadratic equations

Let Q.
/ be a non-degenerate quadratic form on Fd Given any y 2 Fd and anysubsetE  Fqd, define the pinned distance set by

Qy.E/D ¹Q.x y/W x 2 Eº:

Chapman et al [10] obtained the following result using Fourier analysis method

Theorem 2.23 ([10]) LetE Fqd,d  2 Suppose that jEj  qdC12 There exists

a subsetE0ofE withjE0j & jEj such that for every y 2 E, one has

jQy.E/j > q=2;

whereQ.x/D x21C


C x2d

As a corollary of our graph theoretic results, we will present another proof ofTheorem 2.23 In fact, we will prove a more general result.

Theorem 2.24 LetQ be any non-degenerate quadratic form on Fd LetE Fd,

d  2 Suppose that jEj  q.d C1/=2 There exists a subsetE0  E with nalityjE0j D 1 o.1//jEj such that for every y 2 E0, one has

cardi-jQy.E/j D 1 o.1//q:

Note that the proof of [10, Theorem 2.3] implies Theorem 2.24 and vice versa

We again relax the condition on jEj  q.d C1/=2tojEj  q.d C1/=2 to simplifythe argument in the proof

Next we will prove the following result on the solvability of system of quadraticequations (or equivalently, the existence of the simplices over finite fields).Theorem 2.25 For any non-degenerate quadratic formQ on Fdand anyE Fd,

we consider the following system ofl 2t
equations:

Theorem 2.26 For any non-degenerate quadratic formQ and any E  Sd.Q/,

we consider the following system ofl 2t
equations:

P ai aj/D ij; ai 2 E; 1  i  t;

in Fqd, whereE is a large subset of Fqd and P 2 FqŒx1; : : : ; xd One can showthat for a large family of non-degenerate polynomials P , if jEj  q.d Ct 1/=2,that system is solvable for at least cq.t2/ possible choices of ij 2 Fq However, inorder to keep this paper concise, we will restrict our discussion only to results onundirected n; d; /-graphs and their applications.

3 Properties of pseudo-random graphs

We shall recall some results on the distribution of edges in n; d; /-graphs Fortwo (not necessarily) disjoint subsets of vertices U; W  V , let e.U; W / be thenumber of ordered pairs u; w/ such that u 2 U , w 2 W , and u; w/ is an edge

of G For a vertex v of G, let N.v/ denote the set of vertices of G adjacent to vand let d.v/ denote its degree Similarly, for a subset U of the vertex set, let

NU.v/D N.v/ \ U and dU.v/D jNU.v/j:

We first recall the following two well-known facts (see, for example, [4])

Theorem 3.1 ([4, Theorem 9.2.4]) Let G D V; E/ be an n; d; /-colored graph.For any subsetU of V , we have

X

v2V

.dU.v/ djU j=n/2< 2jU j:

The following result is an easy corollary of Theorem 3.1

Corollary 3.2 ([4, Corollary 9.2.5]) Let G D V; E/ be an n; d; /-graph Forany two setsB; C  V , we have

ˇˇe.B; C / djBjjC j

Proof It follows from Theorem 3.1 that

ˇˇ

ˇˇX

v2B

NC.v/ d

njBjjC j

ˇˇ

ˇˇ

completing the proof of the lemma

4 Complete bipartite subgraphs – Proof of Theorem 2.2

Let U1; U2be any subsets of V D V G/ For any y1; : : : ; yt 2 U2, let

Sy 1 ;:::;y t.U1/D ¹x 2 U1W x; yi/2 E.G/; 1  i  tº;

Sy 1 ;:::;y t.U1/D jSy 1 ;:::;y t.U1/j:

When Sy 1 ;:::;y k.U1/ 1, we say that the base y1; : : : ; yk/ is extendable to k-starswith roots in U1 In order to make our inductive argument work, we will need thefollowing definition

Definition 4.1 Let f , g, h be any three functions on the same variables We saythat

f D Qo.g; h/

if f D o.g/ when h D o.g/, and f D O.h/ otherwise

Theorem 2.2 for the star K1;t follows immediately from the following estimate.Lemma 4.2 LetGD V; E/ be an n; d; /-graph For any t  1 and two subsets

Proof The proof proceeds by induction The base case t D 1 follows immediatelyfrom Corollary 3.2 and the fact that

:(4.2)

The lemma follows immediately from (4.2), (4.3) and the additivity of the tion Qo

func-We shall now give a full proof of Theorem 2.2 Let U1and U2be any subsets

of V D V G/ For any y1; : : : ; yt 2 U2, let

Similarly, when

Kys

1 ;:::;yk.U1/ 1;

we say that the base y1; : : : ; yk/ is extendable to Ks;t graphs with the s-parts

in U1 Theorem 2.2 follows immediately from the following estimate

Lemma 4.3 LetG D V; E/ be an n; d; /-graph For any t  s  1 and twosubsetsU1; U2 V , we have

Proof The proof proceeds by induction on s We first consider the base case sD 1.Lemma 4.2 for the stars implies that

t

jU1jjU2jtC1

t

 nd

t 2

2tjU1j1 t;which implies that

The base case sD 1 follows Suppose that the claim holds for s 1  1 We showthat it also holds for s Note that

because both sides equal the number of ordered (possibly degenerate) Ks;t inGŒU1; U2 It follows from Lemma 4.2 that

@X

t

jU1j.Sz 1 ;:::;z s 1.U2//t; qt22tjU1js t

1

A:(4.6)

Besides, by induction hypothesis,