60,131 research outputs found
Mixed pairwise cross intersecting families (I)
An -cross intersecting system is a set of non-empty
pairwise cross-intersecting families with and . If an -cross intersecting system contains at least two families which are
cross intersecting freely and at least two families which are cross
intersecting but not freely, then we say that the cross intersecting system is
of mixed type. All previous studies are on non-mixed type, i.e, under the
condition that . In this paper, we study for the first
interesting mixed type, an -cross intersecting system
with , i.e., families and are cross
intersecting freely if and only if . Let denote the maximum sum of sizes of families in an -cross intersecting system. We determine and
characterize all extremal -cross intersecting systems for
. We think that the characterization of maximal cross
intersecting L-initial families and the unimodality of functions in this paper
are interesting in their own, in addition to the extremal result. The most
general condition on is that . This paper provides foundation
work for the solution to the most general condition
-cross -intersecting families via necessary intersection points
Given integers and we call families
-cross
-intersecting if for all , , we have
. We obtain a strong generalisation of
the classic Hilton-Milner theorem on cross intersecting families. In
particular, we determine the maximum of for -cross -intersecting families in the
cases when these are -uniform families or arbitrary subfamilies of
. Only some special cases of these results had been proved
before. We obtain the aforementioned theorems as instances of a more general
result that considers measures of -cross -intersecting families. This
also provides the maximum of for
families of possibly mixed uniformities .Comment: 13 page
On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method
We study the function which denotes the number of maximal
-uniform intersecting families . Improving a
bound of Balogh at al. on , we determine the order of magnitude of
by proving that for any fixed , holds. Our proof is based on Tuza's set pair
approach.
The main idea is to bound the size of the largest possible point set of a
cross-intersecting system. We also introduce and investigate some related
functions and parameters.Comment: 11 page
Multiple cross-intersecting families of signed sets
A k-signed r-set on[n] = {1, ..., n} is an ordered pair (A, f), where A is an r-subset of [n] and f is a function from A to [k]. Families A1, ..., Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj. Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound to one for families of k-signed r-sets on [n]. The main tool developed is an extension of Katona's cyclic permutation argument.peer-reviewe
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