61 research outputs found
An approximate isoperimetric inequality for r-sets
10 pages10 pages10 pages10 pagesWe prove a vertex-isoperimetric inequality for [n]^(r), the set of all r-element subsets of {1,2,...,n}, where x,y \in [n]^(r) are adjacent if |x \Delta y|=2. Namely, if \mathcal{A} \subset [n]^(r) with |\mathcal{A}|=\alpha {n \choose r}, then the vertex-boundary b(\mathcal{A}) satisfies |b(\mathcal{A})| \geq c\sqrt{\frac{n}{r(n-r)}} \alpha(1-\alpha) {n \choose r}, where c is a positive absolute constant. For \alpha bounded away from 0 and 1, this is sharp up to a constant factor (independent of n and r).The research of the first author was supported by EPSRC grant EP/G056730/1; the research of the third author was supported in part by ERC grant 239696 and EPSRC grant EP/G056730/1
Pairwise Intersections and Forbidden Configurations
Let denote the maximum size of a family of subsets of an -element set for which there is no pair of subsets with , , , and . By symmetry we can assume and . We show that is if either or . We also show that is and is . This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest
Diagonally Neighbour Transitive Codes and Frequency Permutation Arrays
Constant composition codes have been proposed as suitable coding schemes to
solve the narrow band and impulse noise problems associated with powerline
communication. In particular, a certain class of constant composition codes
called frequency permutation arrays have been suggested as ideal, in some
sense, for these purposes. In this paper we characterise a family of neighbour
transitive codes in Hamming graphs in which frequency permutation arrays play a
central rode. We also classify all the permutation codes generated by groups in
this family
The early evolution of the H-free process
The H-free process, for some fixed graph H, is the random graph process
defined by starting with an empty graph on n vertices and then adding edges one
at a time, chosen uniformly at random subject to the constraint that no H
subgraph is formed. Let G be the random maximal H-free graph obtained at the
end of the process. When H is strictly 2-balanced, we show that for some c>0,
with high probability as , the minimum degree in G is at least
. This gives new lower bounds for
the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite
graphs with . When H is a complete graph with we show that for some C>0, with high probability the independence number of
G is at most . This gives new lower bounds
for Ramsey numbers R(s,t) for fixed and t large. We also obtain new
bounds for the independence number of G for other graphs H, including the case
when H is a cycle. Our proofs use the differential equations method for random
graph processes to analyse the evolution of the process, and give further
information about the structure of the graphs obtained, including asymptotic
formulae for a broad class of subgraph extension variables.Comment: 36 page
The multicovering radius problem for some types of discrete structures
The covering radius problem is a question in coding theory concerned with
finding the minimum radius such that, given a code that is a subset of an
underlying metric space, balls of radius over its code words cover the
entire metric space. Klapper introduced a code parameter, called the
multicovering radius, which is a generalization of the covering radius. In this
paper, we introduce an analogue of the multicovering radius for permutation
codes (cf. Keevash and Ku, 2006) and for codes of perfect matchings (cf. Aw and
Ku, 2012). We apply probabilistic tools to give some lower bounds on the
multicovering radii of these codes. In the process of obtaining these results,
we also correct an error in the proof of the lower bound of the covering radius
that appeared in Keevash and Ku (2006). We conclude with a discussion of the
multicovering radius problem in an even more general context, which offers room
for further research.Comment: To appear in Designs, Codes and Cryptography (2012
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Hypergraph matchings and designs
We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs
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