Abstract
AbstractA family $$\mathcal {F} \subset \mathcal {P}(n)$$
F
⊂
P
(
n
)
is r-wisek-intersecting if $$|A_1 \cap \dots \cap A_r| \ge k$$
|
A
1
∩
⋯
∩
A
r
|
≥
k
for any $$A_1, \dots , A_r \in \mathcal {F}$$
A
1
,
⋯
,
A
r
∈
F
. It is easily seen that if $$\mathcal {F}$$
F
is r-wise k-intersecting for $$r \ge 2$$
r
≥
2
, $$k \ge 1$$
k
≥
1
then $$|\mathcal {F}| \le 2^{n-1}$$
|
F
|
≤
2
n
-
1
. The problem of determining the maximum size of a family $$\mathcal {F}$$
F
that is both $$r_1$$
r
1
-wise $$k_1$$
k
1
-intersecting and $$r_2$$
r
2
-wise $$k_2$$
k
2
-intersecting was raised in 2019 by Frankl and Kupavskii (Combinatorica 39:1255–1266, 2019). They proved the surprising result that, for $$(r_1,k_1) = (3,1)$$
(
r
1
,
k
1
)
=
(
3
,
1
)
and $$(r_2,k_2) = (2,32)$$
(
r
2
,
k
2
)
=
(
2
,
32
)
then this maximum is at most $$2^{n-2}$$
2
n
-
2
, and conjectured the same holds if $$k_2$$
k
2
is replaced by 3. In this paper we shall not only prove this conjecture but we shall also determine the exact maximum for $$(r_1,k_1) = (3,1)$$
(
r
1
,
k
1
)
=
(
3
,
1
)
and $$(r_2,k_2) = (2,3)$$
(
r
2
,
k
2
)
=
(
2
,
3
)
for all n.
Publisher
Springer Science and Business Media LLC