8,546 research outputs found
Counting Intersecting and Pairs of Cross-Intersecting Families
A family of subsets of is called {\it intersecting} if any
two of its sets intersect. A classical result in extremal combinatorics due to
Erd\H{o}s, Ko, and Rado determines the maximum size of an intersecting family
of -subsets of . In this paper we study the following
problem: how many intersecting families of -subsets of are
there? Improving a result of Balogh, Das, Delcourt, Liu, and Sharifzadeh, we
determine this quantity asymptotically for and
. Moreover, under the same assumptions we also determine
asymptotically the number of {\it non-trivial} intersecting families, that is,
intersecting families for which the intersection of all sets is empty. We
obtain analogous results for pairs of cross-intersecting families
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
Some New Bounds For Cover-Free Families Through Biclique Cover
An cover-free family is a family of subsets of a finite set
such that the intersection of any members of the family contains at least
elements that are not in the union of any other members. The minimum
number of elements for which there exists an with blocks is
denoted by .
In this paper, we show that the value of is equal to the
-biclique covering number of the bipartite graph whose vertices
are all - and -subsets of a -element set, where a -subset is
adjacent to an -subset if their intersection is empty. Next, we introduce
some new bounds for . For instance, we show that for
and
where is a constant satisfies the
well-known bound . Also, we
determine the exact value of for some values of . Finally, we
show that whenever there exists a Hadamard matrix of
order 4d
Cross-intersecting families and primitivity of symmetric systems
Let be a finite set and , the power set of ,
satisfying three conditions: (a) is an ideal in , that is,
if and , then ; (b) For with , if for any
with ; (c) for every . The
pair is called a symmetric system if there is a group
transitively acting on and preserving the ideal . A
family is said to be a
cross--family of if for any and with . We prove that if is a
symmetric system and is a
cross--family of , then where . This generalizes Hilton's theorem on
cross-intersecting families of finite sets, and provides analogs for
cross--intersecting families of finite sets, finite vector spaces and
permutations, etc.
Moreover, the primitivity of symmetric systems is introduced to characterize
the optimal families.Comment: 15 page
On Mubayi's Conjecture and conditionally intersecting sets
Mubayi's Conjecture states that if is a family of -sized
subsets of which, for , satisfies whenever
for all distinct sets , then , with equality occurring only if is the family
of all -sized subsets containing some fixed element. This paper proves that
Mubayi's Conjecture is true for all families that are invariant with respect to
shifting; indeed, these families satisfy a stronger version of Mubayi's
Conjecture. Relevant to the conjecture, we prove a fundamental bijective
duality between -unstable families and -unstable families.
Generalising previous intersecting conditions, we introduce the
-conditionally intersecting condition for families of sets and prove
general results thereon. We conjecture on the size and extremal structures of
families that are -conditionally
intersecting but which are not intersecting, and prove results related to this
conjecture. We prove fundamental theorems on two -conditionally
intersecting families that generalise previous intersecting families, and we
pose an extension of a previous conjecture by Frankl and F\"uredi on
-conditionally intersecting families. Finally, we generalise a
classical result by Erd\H{o}s, Ko and Rado by proving tight upper bounds on the
size of -conditionally intersecting families and by characterising the families that attain these bounds. We extend
this theorem for certain parametres as well as for sufficiently large families
with respect to -conditionally intersecting families
whose members have at most a fixed number
members
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