32,651 research outputs found
On the structure of dense graphs, and other extremal problems
Extremal combinatorics is an area of mathematics populated by problems that are easy to state, yet often difficult to resolve. The typical question in this field is the following: What is the maximum or minimum size of a collection of finite objects (e.g., graphs, finite families of sets) subject to some set of constraints? Despite its apparent simplicity, this question has led to a rather rich body of work. This dissertation consists of several new results in this field.The first two chapters concern structural results for dense graphs, thus justifying the first part of my title. In the first chapter, we prove a stability result for edge-maximal graphs without complete subgraphs of fixed size, answering questions of Tyomkyn and Uzzell. The contents of this chapter are based on joint work with Kamil Popielarz and Julian Sahasrabudhe.The second chapter is about the interplay between minimum degree and chromatic number in graphs which forbid a specific set of `small\u27 graphs as subgraphs. We determine the structure of dense graphs which forbid triangles and cycles of length five. A particular consequence of our work is that such graphs are 3-colorable. This answers questions of Messuti and Schacht, and Oberkampf and Schacht. This chapter is based on joint work with Shoham Letzter.Chapter 3 departs from undirected graphs and enters the domain of directed graphs. Specifically, we address the connection between connectivity and linkedness in tournaments with large minimum out-degree. Making progress on a conjecture of Pokrovskiy, we show that, for any positive integer , any -connected tournament with large enough minimum out-degree is -linked. This chapter is based on joint work with Ant{\\u27o}nio Gir{\~a}o.ArrayThe final chapter leaves the world of graphs entirely and examines a problem in finite set systems.More precisely, we examine an extremal problem on a family of finite sets involving constraints on the possible intersectionsizes these sets may have. Such problems have a long history in extremal combinatorics. In this chapter, we are interested in the maximum number of disjoint pairs a family of sets can have under various restrictions on intersection sizes. We obtain several new results in this direction. The contents of this chapter are based on joint work with Ant{\\u27o}nio Gir{\~a}o
Supersaturation and stability for forbidden subposet problems
We address a supersaturation problem in the context of forbidden subposets. A
family of sets is said to contain the poset if there is an
injection such that implies . The poset on four elements with is
called butterfly. The maximum size of a family
that does not contain a butterfly is as proved by De Bonis, Katona, and
Swanepoel. We prove that if contains
sets, then it has to contain at least copies of the butterfly provided for some positive . We show by a
construction that this is asymptotically tight and for small values of we
show that the minimum number of butterflies contained in is
exactly
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
A Special Class of Almost Disjoint Families
The collection of branches (maximal linearly ordered sets of nodes) of the
tree (ordered by inclusion) forms an almost disjoint
family (of sets of nodes). This family is not maximal -- for example, any level
of the tree is almost disjoint from all of the branches. How many sets must be
added to the family of branches to make it maximal? This question leads to a
series of definitions and results: a set of nodes is {\it off-branch} if it is
almost disjoint from every branch in the tree; an {\it off-branch family} is an
almost disjoint family of off-branch sets; {\frak o}=\min\{|{\Cal O}|: {\Cal
O} is a maximal off-branch family. Results concerning include:
(in ZFC) , and (consistent with ZFC) is not
equal to any of the standard small cardinal invariants , ,
, or . Most of these consistency results use
standard forcing notions -- for example, comes from starting with a model of and
adding -many Cohen reals. Many interesting open questions remain,
though -- for example,
A coding problem for pairs of subsets
Let be an --element finite set, an integer. Suppose that
and are pairs of disjoint -element subsets of
(that is, , , ). Define the distance of these pairs by . This is the
minimum number of elements of one has to move to obtain the other
pair . Let be the maximum size of a family of pairs of
disjoint subsets, such that the distance of any two pairs is at least .
Here we establish a conjecture of Brightwell and Katona concerning an
asymptotic formula for for are fixed and . Also,
we find the exact value of in an infinite number of cases, by using
special difference sets of integers. Finally, the questions discussed above are
put into a more general context and a number of coding theory type problems are
proposed.Comment: 11 pages (minor changes, and new citations added
Core congestion is inherent in hyperbolic networks
We investigate the impact the negative curvature has on the traffic
congestion in large-scale networks. We prove that every Gromov hyperbolic
network admits a core, thus answering in the positive a conjecture by
Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which
is based on the experimental observation by Narayan and Saniee, Physical Review
E, 84 (2011) that real-world networks with small hyperbolicity have a core
congestion. Namely, we prove that for every subset of vertices of a
-hyperbolic graph there exists a vertex of such that the
disk of radius centered at intercepts at least
one half of the total flow between all pairs of vertices of , where the flow
between two vertices is carried by geodesic (or quasi-geodesic)
-paths. A set intercepts the flow between two nodes and if
intersect every shortest path between and . Differently from what
was conjectured by Jonckheere et al., we show that is not (and cannot be)
the center of mass of but is a node close to the median of in the
so-called injective hull of . In case of non-uniform traffic between nodes
of (in this case, the unit flow exists only between certain pairs of nodes
of defined by a commodity graph ), we prove a primal-dual result showing
that for any the size of a -multi-core (i.e., the number
of disks of radius ) intercepting all pairs of is upper bounded by
the maximum number of pairwise -apart pairs of
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