594 research outputs found
Twins in words and long common subsequences in permutations
A large family of words must contain two words that are similar. We
investigate several problems where the measure of similarity is the length of a
common subsequence.
We construct a family of n^{1/3} permutations on n letters, such that LCS of
any two of them is only cn^{1/3}, improving a construction of Beame, Blais, and
Huynh-Ngoc. We relate the problem of constructing many permutations with small
LCS to the twin word problem of Axenovich, Person and Puzynina. In particular,
we show that every word of length n over a k-letter alphabet contains two
disjoint equal subsequences of length cnk^{-2/3}.
Many problems are left open.Comment: 18+epsilon page
The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques
Let be a sequence of natural numbers. For a
graph , let denote the number of colourings of the edges
of with colours such that, for every , the
edges of colour contain no clique of order . Write
to denote the maximum of over all graphs on vertices.
This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it
has been solved only for a very small number of non-trivial cases.
We prove that, for every and , there is a complete
multipartite graph on vertices with . Also, for every we construct a finite
optimisation problem whose maximum is equal to the limit of as tends to infinity. Our final result is a
stability theorem for complete multipartite graphs , describing the
asymptotic structure of such with in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So
Consistent random vertex-orderings of graphs
Given a hereditary graph property , consider distributions of
random orderings of vertices of graphs that are preserved
under isomorphisms and under taking induced subgraphs. We show that for many
properties the only such random orderings are uniform, and give
some examples of non-uniform orderings when they exist
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