17,462 research outputs found
Erdos-Ko-Rado theorems for simplicial complexes
A recent framework for generalizing the Erdos-Ko-Rado Theorem, due to
Holroyd, Spencer, and Talbot, defines the Erdos-Ko-Rado property for a graph in
terms of the graph's independent sets. Since the family of all independent sets
of a graph forms a simplicial complex, it is natural to further generalize the
Erdos-Ko-Rado property to an arbitrary simplicial complex. An advantage of
working in simplicial complexes is the availability of algebraic shifting, a
powerful shifting (compression) technique, which we use to verify a conjecture
of Holroyd and Talbot in the case of sequentially Cohen-Macaulay near-cones.Comment: 14 pages; v2 has minor changes; v3 has further minor changes for
publicatio
Degrees in oriented hypergraphs and sparse Ramsey theory
Let be an -uniform hypergraph. When is it possible to orient the edges
of in such a way that every -set of vertices has some -degree equal
to ? (The -degrees generalise for sets of vertices what in-degree and
out-degree are for single vertices in directed graphs.) Caro and Hansberg asked
if the obvious Hall-type necessary condition is also sufficient.
Our main aim is to show that this is true for large (for given ), but
false in general. Our counterexample is based on a new technique in sparse
Ramsey theory that may be of independent interest.Comment: 20 pages, 3 figure
On the Parameterized Intractability of Monadic Second-Order Logic
One of Courcelle's celebrated results states that if C is a class of graphs
of bounded tree-width, then model-checking for monadic second order logic
(MSO_2) is fixed-parameter tractable (fpt) on C by linear time parameterized
algorithms, where the parameter is the tree-width plus the size of the formula.
An immediate question is whether this is best possible or whether the result
can be extended to classes of unbounded tree-width. In this paper we show that
in terms of tree-width, the theorem cannot be extended much further. More
specifically, we show that if C is a class of graphs which is closed under
colourings and satisfies certain constructibility conditions and is such that
the tree-width of C is not bounded by \log^{84} n then MSO_2-model checking is
not fpt unless SAT can be solved in sub-exponential time. If the tree-width of
C is not poly-logarithmically bounded, then MSO_2-model checking is not fpt
unless all problems in the polynomial-time hierarchy can be solved in
sub-exponential time
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
Intersections of hypergraphs
Given two weighted k-uniform hypergraphs G, H of order n, how much (or
little) can we make them overlap by placing them on the same vertex set? If we
place them at random, how concentrated is the distribution of the intersection?
The aim of this paper is to investigate these questions
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