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p
1
≥
1)
n
(1
−
2
/p
1
)
,
(
p
1
−
r
log
p
1
1)
.
It can be shown by simple computation that the right-hand side of the last
inequality is strictly smaller than
r
+3 if
p
1
n
≤
(1
−
2
/p
1
) log(
p
1
−
2
r/
3+
t
(
r
) for some function
t
(
r
)=
O
(
r/
log
r
) and, in particular, for
p
1
=
r
+1
≤
≥
8. In this case, we have
n
r
+ 2 relevant coordinates. Thus, Theorem
8
applies, yielding that
A
and
B
are balls. This proves the last statement of Theorem
7
.
For the proof of the second statement, note that we have already established
that
A
and
B
are balls of radius 0 or 1. We use Lemma
4
to calculate the sizes of
A
in
B
in the three possible cases. The product
≤
is
z
1
=
i
=
r
+1
p
i
|
A
|·|
B
|
if
A
and
B
are balls of radius 0. The same product is
z
2
=(
r
+1
r
)
i
=
r
+2
p
i
if
one of them is a ball of radius 0 while the other is a ball of radius 1. Finally, the
i
=1
p
i
−
product is
z
3
=(
r
+2
1)
2
i
=
r
+3
p
i
if both families are balls of radius
1. Note that we have
A
=
B
in the first and third cases. Using the condition
p
i
≥
i
=1
p
i
−
r
−
r
+ 1, it is easy to verify that
z
2
<z
1
and
z
3
≤
z
1
. Furthermore, we have
z
3
=
z
1
if and only if
p
i
=
r
+ 1 for all
i
∈
[
r
+ 2]. This completes the proof of
Theorem
7
.
5 Coordinates with
p
i
=2
In many of our results, we had to assume
p
i
>
2 for all coordinates of the size
vector. Here we elaborate on why the coordinates
p
i
= 2 behave differently.
For the simple characterization of the cases of equality in Theorem
1
,the
assumption
k
= 2 is necessary. Here we characterize all maximal cross-intersecting
pairs in the case
k
=2.
Let
p
=(
p
1
,...,p
n
) be a size vector of positive integers with
k
= min
i
p
i
=2
and let
I
=
{
i
∈
[
n
]:
p
i
=2
}
. For any set
W
of functions
I
ₒ
[2], define the
families
A
W
=
{
x
∈
S
p
:
∃
f
∈
W
such that
x
i
=
f
(
i
) for every
i
∈
I
}
,
B
W
=
{
y
∈
S
p
:
∃
f
∈
W
such that
y
i
=
f
(
i
) for every
i
∈
I
}
.
The families
A
W
and
B
W
are cross-intersecting for any
W
. Moreover, if
|
W
|
=
2
|I|−
1
,wehave
2
/
4, so they form a maximal cross-intersecting
pair. Note that these include more examples than just the pairs of families
described in Theorem
1
, provided that
|
A
W
|·|
B
W
|
=
|
S
p
|
>
1.
We claim that all maximal cross-intersecting pairs are of the form constructed
above. To see this, consider a maximal pair
A, B
|
I
|
ↆ
S
p
.Weknowfromthe
proof of Theorem
1
that
x
∈
A
if and only if
x
∈
/
B
, where
x
is defined by
x
i
=(
x
i
+1mod
p
i
) for all
i
∈
[
n
]. Let
j
∈
[
n
] be a coordinate with
p
j
>
2. By
A
holds if and only if
x
the same argument, we also have that
x
∈
∈
/
B
, where
x
i
=
x
i
for
i
and
x
j
=(
x
j
+2mod
p
j
). Thus, both
x
and
x
belong
∈
[
n
]
\{
j
}
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