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A note on distinct differences in -intersecting families
For a family of subsets of , let
be the
collection of all (setwise) differences of . The family
is called a -intersecting family, if for some positive integer
and any two members we have . The
family is simply called intersecting if . Recently, Frankl
proved an upper bound on the size of for the
intersecting families . In this note we extend the result of
Frankl to -intersecting families
Cross-intersecting families of vectors
Given a sequence of positive integers , let
denote the family of all sequences of positive integers
such that for all . Two families of sequences (or vectors),
, are said to be -cross-intersecting if no matter how we
select and , there are at least distinct indices
such that . We determine the maximum value of over all
pairs of - cross-intersecting families and characterize the extremal pairs
for , provided that . The case is
quite different. For this case, we have a conjecture, which we can verify under
additional assumptions. Our results generalize and strengthen several previous
results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and
answers a question of Zhang
Regular Intersecting Families
We call a family of sets intersecting, if any two sets in the family
intersect. In this paper we investigate intersecting families of
-element subsets of such that every element of
lies in the same (or approximately the same) number of members of
. In particular, we show that we can guarantee if and only if .Comment: 15 pages, accepted versio
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