0 7 8 9
6 1 7 8
5 0 2 7
1+ 6 1 3
0 6 5 1+ 9 8
7 1 0 6 5 9
8 7 2 1 0 6
9 8 7 3 2 1
1 9 8 7 1+ 3
3
5
2
1+
6
2 9 8 7
1+ 3 9 8
3 1+ 5 6
5 6 0 1
0 1 2 3
5
7
0
2
1+
7
8
9
0
2
1+
6
1
3
5
1
2
3
1+
1
9
8
7
2
3
1+
5
5 6
0
1
8
6
0
7
8
9
9
7
3 5 2
6
2
3 5 0
9 3 1+ 6 1
8 9 5 0 2
8 6 1 3
3
7 0 2 1+
1+
6 1 3 5
5
0 7 8 9
6
1 9 7 8
0
2 8 9 7
1
2
9
7
8
'"
'"
COM BIN A TOR 1 A L
MATHEMATICS
YEA R
February 1969 - June 19'10
A GENERALIZATION OF SOME GENERALIZATIONS OF SPERNER'S THEOREM
by
G. KATONA
2
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 600.8
MAY 1969
1 This research was partially sponsored by the National Science
Foundation under Grant No. GU-2059.
2 University of North Carolina and Hungarian Academy of Sciences,
Budapest.
l
A GENERALIZATION OF SOME GENERALIZATIONS OF SPERNER'S THEOREM
l
by
2
G. Katona
Mathematical Institute of the Hungarian Academy of Sciences,
Budapest
INTRODUCTION
Sperner proved the following theorem [l] :
a family of subsets of a set
possess the property
S
A. c A.
1
of
(i
J
elements.
n
9: j) ,
then
m
m i f no
of the family form a chain
the sum of the
h
1
... ,
A } be
m
If no two of them
~r[R]I.
,, "
,
the question which is the maximum of
A = {A ,
Let
h+l
L
;
Erd~s answered
i
different elements
The answer [2] is
largest binomial coefficients of order
n.
Kleitman
[3] and Katona [4] independently proved a stronger form of Sperner's
theorem:
and
Let
respectively.
If
S2
be disjoint sets of
A = {A , ... , Am}
=
elements
is a family of subsets of
1
and no two different
and
A., A.
1
J
satisfy the properties
and
or
A. n SeA. n S1
1
then
( n
I
[~]
I'
2 )
l
1
where
J
and
De Bruijn, Tengbergen and
Kruyswijk [5] generalized the original theorem of Sperner in the
1
This research was partially sponsored by the National Science
Foundation under Grant No. GU-2059.
2 This work was done while the author was at the Department of Statistics
of the University of North Carolina at Chapel Hill •.
2
following manner:
on
S
=
{x '
1
Let
.•. , f
.•. , x }
n
f (~) ~ f (~)
i
j
where
ak's
are
If no two different of them satisfy
(for all
k),
m
then
~
(n
n
Z f(x )
k
k=1
of functions satisfying
be integer-valued functions defined
0 ~ fi(~) ~ ~k'
such that
given positive integers.
m
l: a
= k=~
~
M,
where
M
is the number
"\
I
kl.
Recently, Schonheim [61
,)
gave generalizations of both Erd~s's and Kleitman-Katona's results for
integer-valued functions.
The aim of this paper is to give a common
generalization of all these papers in a little more general language.
DEFINITIONS AND THE THEOREM
We will say that a directed graph
G
is a symmetrical chain graph
if
1.
There is a partition of its vertices into disjoint subsets
. ., . ,
(they are called levels) of
and all the directed edges connect a vertex from
k
elements
n
with a vertex from
K.
1
(0 <
- 1. <
- n) •
2.
3.
k.1 = kn-l. (0 ~ i ~ n).
[E.]
2
There is a partition of its vertices into disjoint symmetrical
k
chains, where symmetrical chain is the set of vertices of a directed
path and if the starting point of this directed path is in
the endpoint is in
duced in [5].)
K
..
n-l
K.
1
then
(The notion of symmetrical chain is intro-
I t is easy to see, 2 is a consequence of 3, that is 2 is
not necessary.
Let us consider a set
vertices of the graph
G
S
of
n
elements.
Let its subsets be the
and connect two vertices
A
and
B
3
(from
A to
B ~ A
B) if
subset-graph.
In this case,
k.1
elements,
= 1.
iB - AI
and
K.
This is the so called
is the family of all subsets of
1
thus conditions 1 and 2 easily hold.
i
Condition
3 also holds as it is proved in [5] in a more general case.
Similarly, if we consider the set of integer-valued functions sat0 s f(~)
isfying
S
uk
connect two vertices
f
and
g
(from
f(~)
where
place
as a vertex-set of a graph
f
to
g)
G,
if
then we have
f
g
1:0
except one
L
K.
i.
It is easy to see that condition
1
is in this case the set of
n
functions for which
1 is satisfyed.
L:
k=l
[5] proves that condition 3 also holds.
This is the
function graph.
Now we define the direct sum
graphs.
and
(g
+ H of two symmetrical chain
Its vertices will be the ordered pairs
l'
h)
1
(g2' h 2 »
to
G
h 2)
(g
E
G, h
~
H)
is connected with
only if
or
(g, h)
h1
gl
= h2
g2
=
and
and
gl' g2
h , h
1
2
are connected in
are connected in
G
H
(from
(from
to
g2) .
If
G
is the subset-graph of a set
of a set
S2
(Sl
graph of
S1 u S2'
and
S2
Sl
and
are disjoint), then
H
G
is a subset-graph
+ H is the subset-
The situation is the same in the case of function
graphs; the direct sum of two function graphs is again a function graph.
The generalization of Sperner's theorem (and also of De Bruijn Tengbergen-Kruyswijk's theorem) in this language is the following:
If we have a set
{a , "', am}
1
of vertices of a symmetrical chain-
graph and no two of them are connected with a directed path, then
4
The generalization of Erdos's theorem would sound that if
no
h+l
then
different vertices from
m::;
the sum of the
h
{a 1
, •••
~
largest
,a }
lie in a directed path,
m
(1)
k. 'so
In a direct sum graph, we
1
will use a weaker condition rather than (1):
THEOREM.
KO'
.to '
•
41
...
•
,
III
,
Let
G and
K
(or
l
elements)
n
p
(gl,h 1 ),
... ,
no
h+l
k o'
•
c
•
,
H
k
n
be symmetrical chain-graphs with levels
elements) and
G+H
1
1
1
path in
1
1
w
path in
for some
then
m
lie in a directed
h +1
G
2
p
(of
lie in a directed
w
H
(in this order)
(2)
h.
1
(in this order)
w (1 ::; w ::; h+l),
the number of vertices of the
..:;;
REMARK l.
8
L
W
that is the sum of the
and
,
such that
h. , .. o,h.
1
g. , ... ,g.
Ie,
different ones of them satisfy the conditions
g.
1
....
I f we have a set
re~ective1y.
of vertices of
(gm,h m)
La,
of
n
If
and
h
G and
p
largest levels of G+H,
a
I: k. .t
largest numbers of type
i=O 1 a-i
H
h
are the subset-graphs of the sets
elements, respectively, then condition (2)
8
1
5
becomes
'I
there are no
h+1
A n S1
1
= , ••
A" n S1
Aw+1 n S1 c •••
C
holds for some
A1""'~+1
different subsets
w (1
C
~
'1,+1 n S,;
~
w
in
SluS2
such that I
I
I
A" n S 2 = Aw+1 n S 2 = •••
='1,+1 n S 2
r
I
h+l),
(3)
It is clear if (3) would hold, then
A cAe ... cAe ... c ~
1
2
w
-[1+1
also should hold, that is, in this case we have a weaker condition than
Erd~s's theorem has, but we have the same result.
The relation of this
special case of our theorem to Erd~s's theorem is the same as the r21ation of Kleitman-Katona's result to Sperner's theorem.
REMARK 2,
If we put
h=l
in the preceding example, we obtain
Kleitman-Katona's result.
REMARK 3.
Theorems of Schonheim can be obtained if we use our
theorem for function graphs and we put
with the stronger condition:
f1 <
-
~
f h+ 1
REMARK 4.
Let
Sl
h+l
Let us consider now an other important special case.
f
defined on
S1
such that
There is a directed edge from
p
functions satisfy
~.
G will be a directed path of length
with
or we change condition (1)
be a one-element set, and let the vertices of
"functions"
teger.
for every
no
h=l
instead of
lattice (Fig. 1).
nc
G+H
f
n+l,
0 ::; f
to
g
Let
~
n
G be the
and
only i f
f
is an in-
g = f+l.
Thus,
H be the same graph
is in this case a rectangular
(n+l)x (p+l)
6
p+l
H
n+l
Fig. 1.
G
In this special case, De Bruijn-Tengbergen-Kruyswijk theorem would state
that if we have a set of points of this rectangle no two of them connected with a directed path, then the maximal number of these points is
the length of the maximal diagonal (a diagonal is a set of vertices with
the same co-ordinate-sum) that is,
min(n+l, p+l).
Schonheim's generalization of Erdgs's theorem would state in this
special case that if we have a set of vertices from this rectangle and
no
h+l
(4)
different ones lie in one directed path,
then the maximal number of these points is the sum of the lengths of
the
h
largest different diagonals.
Our theorem says if we exclude the existency of
h+l
different
points lying in a directed path which consists of two straight lines
(Fig. 2) (instead of (4»
w
w + 1
•
h
,
,
2
t
1
~
Fig" 2.
,
•
h
+ 1
•
7
we obtain the same maximum.
LEMMA.
a;
~
(0 -::; i
Let
R
be a graph with vertices
~
0 -::; j
directed edges from
b;
i
and
(i, j)
have a set of vertices of
there are no
More exactly:
h+l
j
are integers),
only to
m
(i, j)
(i, j+l)
where there are
and
(i+l, j).
If we
elements such that
(i ,jl), ... ,(i + ,jh+l)
1
h 1
different vertices
with the property
i
i
i
1
w
< i +
h 1
<
m
~
<
1
(5)
<
jw
jw
jj+l
w (1 s: w s: h+l) ,
for some
then
j
w
sum of the lengths of the
h
largest different diagonals.
PROOFS
PROOF OF THE LEMMA.
~
fixed, and
0
similarly.
Let
j
~
V
b
V.
is called a column.
(i ' j),
O
where
i
is
O
The rows are defined
be the set of vertices satisfying the conditions of
the lemma and denote by
vertices from
The set of vertices
c
t
the number of columns having exactly
Obviously, by (5)
c
a
o
t
if
t
>
h.
t
Thus
+ L
(6)
Let us count in two different ways the number of vertices which are at
least u-th elements of
V
in any column starting from below.
column where the number of elements of
V
is less than
u
In a
we have to
8
count
0,
so counting column by column, we obtain
C
On the other hand, counting rmv by rm""
most
(h-u+l) (b-u+2)
C
u
.
we obtain that this number is at
because we have not to count the first
and in the other rows we can have at most
dition (5).
(h-u+l)~
u + 2c u+ 1 + 3cu+ 2 + ... +
h-u+l
u-l
rows,
such points by con-
Thus, we have the inequality
+ 2c + 1 + 3c + + ... +
u
u 2
(h-u+l)~ ~
(1
(h-u+l)(b-u+2),
~
u s h).
(7)
h
~
We have to maximalize
~
a+l
b-h+2,
i=O
under the conditions (6) and (7).
c.i
then obviously, the optimal solution is
c
~-1
= O.
1
Assume now that
a+l
that there is an optimal solution with
"
optimal solution
c
,
~-3
h
2:
i=l
. .,
~-3'
ic.
,
... , c 1
h
L:
= c'1
1
~
with
u
= a+l,
First we will show
b-h+2.
If we have an
c'
h
<
b-h+2
then
is also an optimal solution, because
c~
l
results that (7) is
c.:
l
~-1 + 2~
C
h
b-h+2.
and the validity of (7) for
=
h-2:
>
c
e,
= c'h-2 + b-h+2 - e,'n
n-2
l
u = h-l:
~
c
'"
i=l
also valid for
u
~-1'
e,
= c'
- 2 (b-h+2-e,' ) ,
n-1
h-1
n
= b-h+2,
h
~,
=
~-1 - 2(b-h+2-~) + 2(b-h+2)
~-1 + 2~ ~ 2(b-h+2),
+ ... + (h-u-l) ch - 2 + (h-u) ~-1 + (h-u+l) ~
= Cu + ... +
+
(h-u)(~_1
(h-u-l)(c~_2
-
+ b-h+l - ~)
2(b-h+2-~»
+ (h-u+l)(b-h+2)
c~ + ... + (h-u-l)c~_2 + (h-u)~_1 + (h-u+l)c~
~
If
l
(h-u+l)(b-u+2).
9
Thus for this special optimal solution we have
c
+ 2c + +
u 1
u
.. + (h-u)c _
h 1
instead of (7).
=
~-1
or
2
1,
••• = c
c.
n-2
If
or
a+1 = b-h+3,
Let us assume that
a+1
>
~
tion
... , C 1'
b-h+2
approprl'ate one:
=
ch - 3
(8).
is fixed)c
with
c.'.
n-l
It is easy to
c _ = 2
h 1
If we have an other optimal solu-
< 2
then we can change for a more
'
l
c.n-2 -- c.'n-2 - 2(2 - c.n-l
)
'
c.n-l -- 2 ,
c~_3 + (2 - ~-1)'
respectively,
b-h+4.
see that there exists an optimal solution for which
(naturally
(8)
then the optimal solution is:
a+1 = b-h+4
if
(l =:; u ~ h-1)
(h-u+l)(h-u)
a+l::; b-h+4,
= O.
1
~
c.n-4 -- c'
'h-4' ... ,
C1
= c'l'
which satisfy
Following this procedure, we obtain that there exists an optimal
solution of form
o
b-h+2,
~
where
v
o
o
2, ... ,c +1
v
~-1
2,
c
o
v
1 or 0,
0,
is determined by (6).
The set of points
(i, j)
satisfying
i+j = k
diagonal of the rectangle and it is denoted by
diagonals are
D
... ,
,
Dk · The
y = [a+b;h+l] •
h
middle
It is easy to
h
0
see, that the number of the h middle diagonals is just
L ic. ,
1
i=l
that is, maximal. It means they are also the h largest diagonals,
y'
because arbitrary
h
D
y+h-l
where
is called the k-th
diagonals satisfy the conditions of the lemma.
The lemma is proved.
PROOF OF THE THEOREM.
By point 3 of definition of the symmetrical
chain-graphs, the vertices of
chains.
Denote by
G'
and
G and
H'
H
are divisible into symmetrical
the graphs which have the same vertex-
10
set as
G
chains.
and
H,
respectively, but they have edges only along these
Thus,
G'
and
It follows that
H'
G' + H'
is a subgraph of
is a subgraph of
to prove the theorem for
G' + H'
G
and
G+H.
instead of
H,
respectively.
So, it is sufficient
G+H.
However,
G' + H'
consists of rectangular lattices and condition (2) means simply condition
(5) for every such rectangle.
We know that an optimal set of points in every rectangle is the
union of the
h
middle diagonals.
G+H,
in the following manner.
some
i.
Define the levels of
(g,h)
E
M
iff
j
g
E
G' + H',
K., h
L..
E
1
or
for
J-1
By the definition of the direct sum, it is easy to see that
M. 's satisfy the 1 point of the definition of a symmetrical chain
J
graph.
where
The
z
h
middle levels of
[n+p-h+l]
=
G' + H'
are
We will show that the union of the
2'
diagonals for all the rectangles is just the
G'
M,
z •.. , Mz+h _. 1 '
h
h
middle
middle levels in
+ H'.
First we verify that an element of the
rectangle is an element of the
h
h
middle diagonals in a
middle levels in
G' + H'.
Let us
consider a fixed rectangle which is a direct sum of two symmetrical
chains from
G'
respectively.
thus
i+a
=
and
If
n-i
H'
go
E
with vertices
K ,
i
then by the symmetricity
n
and
ga
h ' ... , h ,
o
b
E
K _
n i
and
or
i
(Obviously,
go' ... , ga
and
a
n-a
=2
have the same parity.)
(9)
Similarly, i f
h
O
E
L.
J
then
j
p-b
2
(10)
11
is in
D,
in one of the
r
h
largest diagonal of
the rectangle, then
+ .t
k
further,
gk
E
Ki +k ,
h
l
E
Lj+.t,
r,
(11)
thus
or using
(9), (10) and (11)
(12)
Since
~
= [a+b-h+l
2
]
y
z+h-1,
~
s
~
y+h-l,
z
thus
(gk,h.t)
= [n+p-h+l]
-2
-< n+p-a-b
-2
+ r
is in one of the
h
middle
G' + H' .
Conversely, let
~
$
and (12) means that
levels of
z
r
z+h-l.
(g,h)
(g,h)
be an element of
where
M,
s
is contained by one rectangular which is a direct
sum of two symmetrical chains, say,
go' •.. , ga
and
h ' ••• , h
O
b
.
Then, by (9) and (10)
h
o
E L
p-b
2
If
(g,h)
r = k+.t
(gk,h.t) then
p-b
n-a
s - --2
[a+b-h+l]
2
(g,h)
2
n-a
p-b
-2-+ 2 + k + .t = s,
the following inequality holds:
n+p-a-b
= z 2
~
r
~
[a+b-h+l] + n-L
2
is really an element of a diagonal from the
Thus we proved that the points of the
optimal set.
that is, for
For the union of
h
h
h
middle ones.
middle levels form an
arbitrarily chosen levels of
the conditions of the theorem are satisfied, so the
h
G'+H',
middle levels
12
must be the
h
largest ones (but there may be different
the same size-sum).
a
~
i=O
k. l
~
.,
a-~
The number of elements in
M
The proof is completed.
levels with
is obviously
thus the optimal number is the sum of the
ones of these numbers.
h
h
largest
REFERENCES
1.
E. SPERNER,
Ein Satz tiber Untermengen einer endlichen Menge,
Math. Z. 27 (1928), 544-548.
2.
"
P. ERDOS,
On a lemma of Littlewood and Offord,
Bull. Amer. Math. Soc. 51 (1945), 898 -902.
3.
D. KLEITMAN,
On a lemma of Littlewood and Offord on the distri-
bution of certain sums,
Math. Z. 90 (1965), 251-259.
4.
G. KATONA,
On a conjecture of Erdgs and a stronger form of
Sperner's theorem,
Studia Sci. Math. Hungar. 1 (1966), 59-63.
5.
N.G. DE BRUIJN, CA. VAN E. TENGBERGEN and D. KRUYSWIJK,
On the
set of divisors of a number,
Nieuw Arch. Wiskunde (2), 23 (1949-51), 191-193 .
6.
..
J. SCHONHEIM,
A generalization of results of P. Erdgs, G. Katona
and D. Kleitman concerning Sperner's theorem,
J. Combinatorial Theory (to Appear).