15,282 research outputs found
Strong stability of 3-wise -intersecting families
Let be a family of subsets of an -element set. The family
is called -wise -intersecting if the intersection of any
three subsets in is of size at least . For a real number
we define the measure of the family by the sum of
over all . For example, if
consists of all subsets containing a fixed -element set, then it is a
-wise -intersecting family with the measure .
For a given , by choosing sufficiently large, the following
holds for all with . If is a
-wise -intersecting family with the measure at least
, then satisfies one of (i) and (ii): (i)
every subset in contains a fixed -element set, (ii) every
subset in contains at least elements from a fixed
-element set
Non-trivial 3-wise intersecting uniform families
A family of -element subsets of an -element set is called 3-wise
intersecting if any three members in the family have non-empty intersection. We
determine the maximum size of such families exactly or asymptotically. One of
our results shows that for every there exists such that if
and then the maximum size
is .Comment: 12 page
Dual virtual element method for discrete fractures networks
Discrete fracture networks is a key ingredient in the simulation of physical
processes which involve fluid flow in the underground, when the surrounding
rock matrix is considered impervious. In this paper we present two different
models to compute the pressure field and Darcy velocity in the system. The
first allows a normal flow out of a fracture at the intersections, while the
second grants also a tangential flow along the intersections. For the numerical
discretization, we use the mixed virtual finite element method as it is known
to handle grid elements of, almost, any arbitrary shape. The flexibility of the
discretization allows us to loosen the requirements on grid construction, and
thus significantly simplify the flow discretization compared to traditional
discrete fracture network models. A coarsening algorithm, from the algebraic
multigrid literature, is also considered to further speed up the computation.
The performance of the method is validated by numerical experiments
High dimensional Hoffman bound and applications in extremal combinatorics
One powerful method for upper-bounding the largest independent set in a graph
is the Hoffman bound, which gives an upper bound on the largest independent set
of a graph in terms of its eigenvalues. It is easily seen that the Hoffman
bound is sharp on the tensor power of a graph whenever it is sharp for the
original graph.
In this paper, we introduce the related problem of upper-bounding independent
sets in tensor powers of hypergraphs. We show that many of the prominent open
problems in extremal combinatorics, such as the Tur\'an problem for
(hyper-)graphs, can be encoded as special cases of this problem. We also give a
new generalization of the Hoffman bound for hypergraphs which is sharp for the
tensor power of a hypergraph whenever it is sharp for the original hypergraph.
As an application of our Hoffman bound, we make progress on the problem of
Frankl on families of sets without extended triangles from 1990. We show that
if then the extremal family is the star,
i.e. the family of all sets that contains a given element. This covers the
entire range in which the star is extremal. As another application, we provide
spectral proofs for Mantel's theorem on triangle-free graphs and for
Frankl-Tokushige theorem on -wise intersecting families
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