JOURNAL
OF COMBINATORIAL
THEORY,
series A 45, 157-161 (1987)
Note
On Subsets
of Abelian
Arithmetic
Groups with
Progression
No 3-Term
P. FRANKL*
C.N.R.S.,
Paris,
France
R. L. GRAHAM
600 Mountain
AT&T Bell Laboratories
Avenue, Murray
Hill, New Jersey
07974
AND
V. R~DL*
FJFI
CVUT,
Communicated
Prague,
Czechoslovakia
by the Managing
Editors
Received October 25, 1986
A short proof of the following result of Brown and Buhler is given: For any E > 0
there exists n, = no(E) such that if A is an abelian group of odd order IAl > no and
BG A with IBI >&IAI. then B must contain three distinct elements X, y, z satisfying
x + y = 22.
0 1987 Academic
Press, Inc.
1. INTRODUCTION
Let N denote the set of positive integers, and for n E N, let [n] denote
the set { 1, 2,..., n}. A well-known theorem of Roth [R] asserts that if P s N
contains no 3-term arithmetic progression, then P has upper density zero.
That is, for every E> 0, 1P n [n] 1< sn holds for all sufficiently large n.
Brown and Buhler [BBl ] proved the following generalization of Roth’s
result.
* This work was performed while the authors were visiting AT&T Bell Laboratories.
157
0097-3165187 S3.00
Copyright 0 1987 by Academic Press, Inc.
All rights of reproduction
in any form reserved.
158
FRANKL,GRAHAM,ANDRijDL
THEOREM 1. For every E>O there exists n,=q,(E)
with the following
property. Suppose A is an abelian group of odd order, 1Al > n,. Then every
subset Bc A with IBI > EIAI contains three distinct elements x, y, z with
x+ y=2z.
For a finite set X, define (f):={FsX:
(I;[ =k}. A family FE(~) is
called a k-graph. It is called linear if (Fn Cl < 1 holds for all distinct
F, G E F. Three distinct edges, F, G, H of a linear k-graph are said to form a
triangle if the three intersections F n G, G n H, H n F are all non-empty
and distinct.
THEOREM 2 (Ruzsa-Szemeredi
[RS] ). Suppose that F is a Zinear
3-graph on n vertices which contains no triangle. Then IFI = o(n’).
For a simple proof of Theorem 2, see [EFR].
Theorem 1 follows easily from Theorem 2.
2. PROOF OF THEOREM
Here we show that
1
Suppose A is an abelian group of odd order and BE A contains no
three distinct elements x, y, z with x + y = 22. Define X= A x [3] to be the
3(Al-element set with general element (a, i), a E A, 1 6 i < 3. Now define a
3-graph F as
F:={{(a,l),(a+b,2),(a+26,3)}:a~A,b~B}.
Clearly, IFI = 1Al 1BI. Also, F is linear since any two elements of an edge
uniquely determine the edge.
Suppose now to the contrary that F contains a triangle, say
{ (ai, I), (ai + bi, 2), (ai + 2bi, 3)},
i = 1, 2, 3.
By symmetry, we may assume that
aI = a2,
a,+b,=a,+b,,
a2+2b2=a,+2b,.
However, these equations imply
2b,-2b,=a,-a,=a,-a,=b,-b,,
i.e.,
2b, = b, + b,.
By the choice of B, this implies b, = b2 = b3 and thus, a, = a2 = a3, a contradiction. Thus, F contains no triangle.
ABELIAN GROUPS WITH NO j-TERM
PROGRESSION
159
Hence, by Theorem 2,
IFI = IAl 14 = 4lAl’h
i.e.,
I4 =4lAl)
as desired.
Remark. The same proof can be used in the case when A is a d-dimensional affine space over GF(2’), t > 2. For the definition of edges in the
proof, one replaces a + 2b by a + yb where y # 0, 1 is an arbitrary element
of GF(2’). The conclusion then becomes: B contains three points on a line.
3. SOMELOWER BOUNDS
The most important special cases of Theorem 1 are when A is a cyclic
group (corresponding to Roth’s theorem) and when A is an afline space
A(d, q) of dimension d over GF(q).
In both cases, stronger theorems are known. Szemeredi’s theorem [S]
asserts that sets with positive upper density contain arithmetic progressions
of arbitrary length, while a recent result of Furstenberg and Katznelson
[FK] implies that for any E> 0 and any prime power q there exists do =
do(t, q) so that the following is true: Every subset Bc A(d, q) with IBI >
Eqd,d> do, contains all the points of some line in A(d, q).
In view of [BB2] this implies the same statement if we replace lines by
planes, spaces, etc.
Let a,(d) denote the maximum of )BI where Bs A(d, q) contains no line.
In the case of the integers, Behrend [B] showed that for every S > 0 and
n > rr0(6) there exists B c [n] with IBI > rr’ -’ so that B contains no 3-term
arithmetic progression. We do not know if the corresponding statement
holds for afline spaces.
PROBLEM.
Is it true that for every 6 > 0, q 2 3 and d> d,(6, q), there
exists B c A(d, q) which contains no line and satisfies I BI > (q - 8)‘.
It is easy to construct such a B with IBI = (q - 1)“; simply take
B= ((bl,..., b,): biEGF(q)\{O},
i= l,..., d}.
To improve on this bound note that if x E GF(q) is fixed and {(by),..., b$)),
i=l >..*,q} forms a line then the sets Fi = {j: bj’) = x} form a sunflower of
size q, that is, F, n . . . nF,=FjnFj,
holds for all ldj<j’Qq.
Let f&d, r) denote the maximum of IFI where F E ( ‘:I), and F contains
no sunflower of size q.
160
FRANKL,GRAHAM,ANDRGDL
PROPOSITION.
For all positive integrs n, r, d, one has
a,(n) 2 f,(n, r)(q - 1)” ~’
and
a& nd) 2 a&n Y’.
(2)
Proof Let F G ([;I) be a family without sunflowers of size q which
satisfies IFI =fy(n, r).
Fix an element XE GF(q). For b= (b,,..., b,)~A(n, q), define F(b) :=
{j: b]=x} and
B:= (bEA(n,
q): F(b)EF}
Then B contains no line. To prove the second assertion, one simply notes
that if B contains no line then
B@ ... @BsA(n,
q)@ ... @A(n,q)=A(dn,
q)
contains no line either.
If we knew the value of f,(n, r), then probably we could get fairly good
lower bounds on a,(n).
Although this problem goes back to ErdGs and Rado [ER], very little is
known about f&n, r).
For q odd, n = 2q and r = 2, one can take two disjoint complete graphs
on q vertices each. This shows f,(2q, 2) 2 q(q - 1). Actually one has
equality, but we do not need it. Using (1) we obtain
a,(2dq)a(q-
1)2dq
Using the fact that there is a collection of 300 6-element subsets of [18]
without a sunflower of size three, one obtains a3( 18) > 300. 1212 and thus
a,(d) 2 (2.179)d for d> d,.
REFERENCES
Cnl
[BBl]
[BB2]
[EFR]
F. A. BEHREND, On sets of integers which contain no three elements in arithmetic
progression,
Proc. Nat. Acad. Sci. 23 (1946), 331-332.
T. C. BROWN AND J. P. BUHLER, A density version of a geometric
Ramsey theorem,
J. Combin. Theory A 32 (1982), 2&34.
T. C. BROWN AND J. P. BUHLER, Lines imply spaces in density Ramsey Theory,
J. Combin. Theory A 36 (1984), 214-220.
P. ER&s, P. FRANKL, AND V. R~~DL, The asymptotic
number of graphs not containing a fixed subgraph
and a problem
for hypergraphs
having no exponent,
Graphs
and Combine 2 (1986)
113-121.
ABELIAN
WI
[FK]
CR1
WI
PI
GROUPS
WITH
NO
j-TERM
PROGRESSION
161
P. ERD& AND R. RAW, Intersection theorems for systems of sets, J. London Mati
Sm. 35 (1960), 85-90.
H. F~JRSTENBERG
AND Y. KATZNELSON, An ergodic Szemerkdi theorem for P-systems
and combinatorial theory, J. Anal. Math. 45 (1985), 117-168.
K. F. ROTH, On certain sets of integers, J. London Math. Sot. 2.8 (1953), 104-109.
I. Z. RUZSA AND E. SZEMER~DI,Triple systems with no six points carrying three
triangles, Colloq. Math. Sot. Junos Bolyai 18 (1978), 939-945.
E. SZEMER~DI, On sets of integers containing no k elements in arithmetic
progression, Acta A&h. 27 (1975) 199-245.