29,220 research outputs found
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
Most Probably Intersecting Families of Subsets
Let F be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality vertical bar F vertical bar <= 2(n-1). Suppose that vertical bar F vertical bar = 2(n-1) + i. Choose the members of F independently with probability p (delete them with probability 1 - p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing F appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all [n/2]-element subsets. We determine the most probably inclusion-free family too, when the number of members is (n([n/2])) + 1
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
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