6 research outputs found
Cross-intersecting non-empty uniform subfamilies of hereditary families
A set -intersects a set if and have at least common
elements. A set of sets is called a family. Two families and
are cross--intersecting if each set in
-intersects each set in . A family is hereditary
if for each set in , all the subsets of are in
. The th level of , denoted by
, is the family of -element sets in . A set
in is a base of if for each set in
, is not a proper subset of . Let denote
the size of a smallest base of . We show that for any integers
, , and with , there exists an integer
such that the following holds for any hereditary family
with . If is a
non-empty subfamily of , is a non-empty
subfamily of , and are
cross--intersecting, and is maximum under
the given conditions, then for some set in with , either and ,
or , , , and . This was conjectured by the author for and generalizes well-known
results for the case where is a power set.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1805.0524
Cross-intersecting non-empty uniform subfamilies of hereditary families
Two families A and B of sets are cross-t-intersecting if each set
in A intersects each set in B in at least t elements. A family H is
hereditary if for each set A in H, all the subsets of A are in H. Let
H(r) denote the family of r-element sets in H. We show that for
any integers t, r, and s with 1 ≤ t ≤ r ≤ s, there exists an integer
c(r, s, t) such that the following holds for any hereditary family
H whose maximal sets are of size at least c(r, s, t). If A is a nonempty subfamily of H(r)
, B is a non-empty subfamily of H(s)
, A
and B are cross-t-intersecting, and |A| + |B| is maximum under
the given conditions, then for some set I in H with t ≤ |I| ≤ r,
either A = {A ∈ H(r)
: I ⊆ A} and B = {B ∈ H(s)
: |B ∩ I| ≥ t}, or
r = s, t < |I|, A = {A ∈ H(r)
: |A ∩ I| ≥ t}, and B = {B ∈ H(s)
: I ⊆
B}. We give c(r, s, t) explicitly. The result was conjectured by the
author for t = 1 and generalizes well-known results for the case
where H is a power set.peer-reviewe