2,258 research outputs found
Proof of a conjecture on induced subgraphs of Ramsey graphs
An n-vertex graph is called C-Ramsey if it has no clique or independent set
of size C log n. All known constructions of Ramsey graphs involve randomness in
an essential way, and there is an ongoing line of research towards showing that
in fact all Ramsey graphs must obey certain "richness" properties
characteristic of random graphs. More than 25 years ago, Erd\H{o}s, Faudree and
S\'{o}s conjectured that in any C-Ramsey graph there are
induced subgraphs, no pair of which have the same
numbers of vertices and edges. Improving on earlier results of Alon, Balogh,
Kostochka and Samotij, in this paper we prove this conjecture
The random k-matching-free process
Let be a graph property which is preserved by removal of edges,
and consider the random graph process that starts with the empty -vertex
graph and then adds edges one-by-one, each chosen uniformly at random subject
to the constraint that is not violated. These types of random
processes have been the subject of extensive research over the last 20 years,
having striking applications in extremal combinatorics, and leading to the
discovery of important probabilistic tools. In this paper we consider the
-matching-free process, where is the property of not
containing a matching of size . We are able to analyse the behaviour of this
process for a wide range of values of ; in particular we prove that if
or if then this process is likely to
terminate in a -matching-free graph with the maximum possible number of
edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds
on are essentially best possible, and we make a first step towards
understanding the behaviour of the process in the intermediate regime
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