On
shifted
intersecting
fa:rr:tilies
"VV'"ith
r e s p e c t to p o s e t s
Morimasa Tsuchiya
Department of Mathematical Sciences
Tokai University
Hiratsuka, Kanagawa 259-12, JAPAN
In this paper, we show that for a shifted complex :Jr
Abstract.
~ 2P
with respect to a poset P with minimum element 0 and an intersecting
subfamily ~
~
~
#~
:Jr,
#{FE:Jr;OEF}.
We denote the set {1,2, .. ,n} by [n], the family of all subsets of a
set X by 2x ,
#F denotes the number of elements of a set F.
be a family of subsets of [n], i.e.,
E
:Jr,
Fi () F j
'*
=
{F 1, .. ,F m } where F1
Fm
A family g, is intersecting if for every F i ,
are distinct subsets of [n].
Fj
:Jr
Let :Jr
For families ~, g,
0.
'*
cross-intersecting if G () F
0 for V G
E
family :Jr ~ 2[n] is called a complex if G S;;; F
~ and :Jr are
S;;;
2[n],
~
and V F
E
E
g, implies G
:Jr.
A
E
:Jr.
We already know the following results. For an intersecting family :Jr S;;;
2[n J , #:Jr ~ 2n- 1 ([1]) and for a complex :Jr S;;; 2[n] and cross-intersecting
subfamilies ~, Je ~
For F, G
with x
~
~
:Jr,
#~
+ #Je
~
# g, ([4]).
[n], if there exists a one-to-one mapping
I(x) for each x
E
F, then we write F
Australasian Journal of Combinatorics
~ ( 1992
~
G.
I:
:Jr
L pp. 53-58
F --> G
~
2[n]
~
~
is V-hereditary if G
FE
implies G
;}
E
V.Chvatal introduced
;}.
this notion and proved the next result.
Theorem A (1974 [2]).
Let;} ~ 2[n] be a V-hereditary family and ~
be an intersecting subfamily of ;}.
Then # ~ ~ #{ FE;} ; 1 E F }.
•
H.Era extended the notion of V.Chvatal and also showed the following
result.
Let P be a finite ranked poset with the minimum element O.
~
For F, G
x
~
P, if there exists a one-to-one mapping f: F --> G with
f(x) in P or x and f(x) are incomparable for each x
~p
we write F
implies G
E
G.
;}
~
2
P
is P-hereditary if G
E
F, then
~p
FE;}
;}.
Let P be a finite ranked poset with the minimum
Theorem B ([3]).
~ 2 P be a P-hereditary family.
element 0 and ;}
subfamily i:§ of:f,
For an intersecting
#~ ~#{FE;} ;OEF} . •
Let P be a finite poset with the minimum element O. For a family
;} ~ 2P and a ~ (3 in P, we define
S a,/3(F)
=
(F-{(3})u{a} if ai-F, (3EF,(F-{(3})U{a}i-;}
{
F
for each F E ; } and Sa,/3(;})
=#:J
and
if ;}
otherwise
= {Sa,/3(F); FE:J}.
is complex and intersecting,
Then #Sa,/3(;})
then Sa, /3 (;})
is also
complex and intersecting.'
Proposition 1.
complex,
For a finite poset P and
a, ~ (3 in
P,
if ;}
~ 2 P is
then Sa, /3 (;}) is also complex.
Proof. We suppose that there exist G, F
Sa,/3(;}) and G
i-
Sa,/3(;})'
Case 1. FE:;}
54
~
P such that G
~
F
E
Since Y is complex, G
¢.
Y
and
(3
F.
E
If
E
Y.
a
E
¢. G, (3
So a
F, then (G-{(3})u{a}
Y
and
~
(G-{ (3}) u {a}
~
which
F,
If a ¢. F, then (F - { (3 }) u { a }
contradicts the property that g; is complex.
E
G, (G-{ (3)) u { a}
E
(F -{ (3}) u {a},
which
contradicts
the
property that g, is complex.
Case 2. F ¢. g; .
Then a
a
F, (3
E
¢. G, !3
E
G'
G and (G-{!3})u{a}
If G
contradiction.
= (G-{a})u{!3}
= G E Sa,(3(g;),
¢. Y, then
g;.
E
Since G'n{a,!3}
which is a contradiction.
(F -{ a}) u {!3} and
(G'-{!3})u{a}
= {.B},
•
~ !3 in P,
if g;
~ 2 P is
Sa,(3(g;) such that GnF
E
a
Thus there exists HEY such that Sa. (3 (H)
=I F. By the definition of (a, !3)-shifting, H = (F-{ a}) u {(3}
E
F and
!3
¢. F.
0, which is a contradiction.
Since
~
F, which is a
Since Y is intersecting, both of G and F do not belong to Y.
= F and H
Y,
i.
Y, then
E
then Sa, (3 (Y) is also intersecting.
We assume that F ¢. g;.
E
If G
So G
(G-{ a}) u {!3}
Proof. We suppose that there exist G , F
= O.
Y.
E
¢. Y.
For a finite poset P and a
Proposition 2.
intersecting,
¢. F and (F-{a})U{!3}
Sa, (3(G)
( a, !3 )-shifting.
Then
«F-{ a })n(G-{ (3}»
({a}n{(3})
Thus G
u
=(F-{a})n(G-{!3}) = 0,
g;
n
({(3}n(G-{!3}»
E
E
by
G and F n G
G and
the
a
=I
¢. G.
definition
of
«G-{!3})u{a})
u
«F -{ a}) n { a })
u
contradicting the fact that Y
•
A family g; is shifted if Sa,(3(Y)
P.
E
«F-{a))u{!3})
is an intersecting family.
< (3 in
:J, !3
E
(G-{(3})u{a}
G,
=
¢. :J, then a
If G
=
g; for all a, !3 such that a
We obtain the following result which is concerned with
shifted complexes and intersecting families.
55
Let P be a finite poset with the minimum element 0 and
Theorem 3.
S
2
S;;;
#~ ~
P
For an intersecting subfamily ~ of S,
be a shifted complex.
#{ FES; OEF}.
=
Proof. Let :reO)
Proposition 2, we can assume that
~...
family
H
E ~},
= {H ; H S;;; :3 G
~....
E
= { FE S
{F-{O}; OEFE fJ } and So
~
is shifted.
that is, if G
In the following we show that ~...
By
Then we define the
~
E
; OiF}.
and H
G,
S;;;
then
= {H ; H s;;: :3 G E ~} is a
shifted complex.
Suppose that ~.. is not a shifted complex.
and
P
E
H
(H- {/3})v{a}
that H
~
is
i
shifted.
~ ...
a
;;;
(H - {/3}) V { a}
E
H n { a, /3 }
/3 ,
= {/3}, then (G -
(H-{/3})V{a}
{/3}) V { a}
~ .. ,
S;;;
E
H
:3G
E
E ~}
is a shifted complex and ~
~.
Since ~ .. is shifted,
~... Let ~ 0 == { G E ~ ; 0 i G } and '{5
~
For C
0.
#~
'f3 and G
E
For
'{5+
~o' since 0
By the fact that ~
0'.
=
=
E
~ #~o.
i
#S(O).
==
i
0
= { C E fJ 0;
Since ~o and So- 'f3 are cross-intersecting,
and therefore #'f3
S;;;
g;.
#~ - #~o + #'{5
Proposition 4.
::/. {/3},
G E ~,
==
=
c i
0,
#'f3+
0'}.
~
=#'{5.
({O}vC)nG
~.
#S(O).
So 'B + n ~
'f3 +
S;;;
S,
•
Let P be a finite poset with the minimum element O.
56
E
+ #(So- '(3) ;;;;;; #fJ o
G and enG
#«~ - ~o)U '(5+)
S;;;
Therefore
:3 G E ~ 0' C n G ==
#~o
{O} V
~..
H implies RU{O}
={CV{O};C E 'f3},
is intersecting,
S;;;
For Y H
Since every element of (~- ~ 0) U '{5 + contains 0 and
;;;;;;
~,
E
which is a contradiction.
= {H ; H S;;; :3 G
c
~ because
E
(G - {/3}) V {a}
without loss of generality we can assume that ~..
~ .. -~,
and
~ such
If G n { a , /3}
So for ~ .. (O) = {G-{O}; OEGE~ .. }, #~ .. (O)
S.
/3
{ /3}
(G- {/3})v{a}
S;;;
~ .. , which is a contradiction.
Since (H - {/3}) V {a}
G.
E
Thus ~...
that
By definition of ~ .. , there exists G E
Since
(H - { /3 }) V { a} E
then a, /3
such
If G n { a ,/3}
G.
S;;;
~..
E
Then there exist a,
If
g. ,
2 P is a P-hereditary family,
Proof. We assume that G ,
2
P
then g. is a shifted complex.
and G ,
J F E: g..
Since the mapping
~p
f from G to F such that f(x) = x is a one-to-one mapping, G
By the property that g. is a P- hereditary family,
G
g..
E:
F.
Thus g. is
complex.
that a .f3
E: P
and a
and (F-{/3})V{a}
i
/3 and
~
g..
/3 such
F n { a, /3} = {/3}
Then there exist a and
We assume that g. is not shifted.
F
E:
g. such that
We define the mapping f from (F-{/3})V{a}
to F as follows:
f(x) =
Since a
if x -=I a
X
{
~
/3 if x = a.
/3 in P, x
~
f(x) for \Ix
E:
~p
a one-to-one mapping and (F-{/3})v{a}
g. is a P-hereditary family, (F-{ /3}) u {a}
E:
Thus f is
(F-{/3})v{a}.
F.
By the property that
g., which is a contradiction .
•
By Proposition 4 and Theorem 3, we also obtain Theorem B.
the converse of Proposition 4 does not hold.
=
of Figure 1, :J
shifted
complex.
P-hereditary.
However
For example, for the poset
{{O,1,2}, {O,l}, {O,2}, {1,2}, {a}, {I}, {2}} is a
Since
{3}
~P
{2}
and
{3}
i
g.,
g.
is
not
So we do not obtain Theorem 3 from Theorem B.
We can easily see that
,Cj
is a V-hereditary family if and only if g.
is a shifted family with respect to a linear order set.
Let P be a poset
with the minimum element and I(P) be a liner extension of P.
If g. is
a shifted family with respect to I(P), then g. is a shifted family with
respect to P.
So we also obtain Theorem; A by Theorem 3.
converse does not hold.
For example, :J
= {{O,1,2},
But the
{O,3,4}} is a
shifted family with respect to the poset of Figure 1 and is not a shifted
family with respect to the liner extension
57
°
~
1 ~ 2 ~ 3 ~ 4.
Figure 1.
References.
[1] I.,Anderson, Combinatorics of finite sets,Oxford Univ. Press, (1987).
[2] V.,Chvatal, Intersecting families oj edges in hyper graphs having
the heredita ry property ,Hypergraph Seminar, LNM 411,Springer (1974)
61-66.
[3] H.,Era, A comment on a Chvatal's conjecture,
[4] L,Marica and J.,Schonheim,
preprint.
Differences of sets and a problem of
Graham, Can. Math. Bull. 12 (1969) 635-637.
58