31,737 research outputs found
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
Density theorems for intersection graphs of t-monotone curves
A curve \gamma in the plane is t-monotone if its interior has at most t-1
vertical tangent points. A family of t-monotone curves F is \emph{simple} if
any two members intersect at most once. It is shown that if F is a simple
family of n t-monotone curves with at least \epsilon n^2 intersecting pairs
(disjoint pairs), then there exists two subfamilies F_1,F_2 \subset F of size
\delta n each, such that every curve in F_1 intersects (is disjoint to) every
curve in F_2, where \delta depends only on \epsilon. We apply these results to
find pairwise disjoint edges in simple topological graphs
Density theorems for bipartite graphs and related Ramsey-type results
In this paper, we present several density-type theorems which show how to
find a copy of a sparse bipartite graph in a graph of positive density. Our
results imply several new bounds for classical problems in graph Ramsey theory
and improve and generalize earlier results of various researchers. The proofs
combine probabilistic arguments with some combinatorial ideas. In addition,
these techniques can be used to study properties of graphs with a forbidden
induced subgraph, edge intersection patterns in topological graphs, and to
obtain several other Ramsey-type statements
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
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 New Framework for Network Disruption
Traditional network disruption approaches focus on disconnecting or
lengthening paths in the network. We present a new framework for network
disruption that attempts to reroute flow through critical vertices via vertex
deletion, under the assumption that this will render those vertices vulnerable
to future attacks. We define the load on a critical vertex to be the number of
paths in the network that must flow through the vertex. We present
graph-theoretic and computational techniques to maximize this load, firstly by
removing either a single vertex from the network, secondly by removing a subset
of vertices.Comment: Submitted for peer review on September 13, 201
Largest minimal inversion-complete and pair-complete sets of permutations
We solve two related extremal problems in the theory of permutations. A set
of permutations of the integers 1 to is inversion-complete (resp.,
pair-complete) if for every inversion , where 1 \le i \textless{} j \le
n, (resp., for every pair , where ) there exists a
permutation in~ where is before~. It is minimally inversion-complete
if in addition no proper subset of~ is inversion-complete; and similarly for
pair-completeness. The problems we consider are to determine the maximum
cardinality of a minimal inversion-complete set of permutations, and that of a
minimal pair-complete set of permutations. The latter problem arises in the
determination of the Carath\'eodory numbers for certain abstract convexity
structures on the -dimensional real and integer vector spaces. Using
Mantel's Theorem on the maximum number of edges in a triangle-free graph, we
determine these two maximum cardinalities and we present a complete description
of the optimal sets of permutations for each problem. Perhaps surprisingly
(since there are twice as many pairs to cover as inversions), these two maximum
cardinalities coincide whenever
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