8,906 research outputs found
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Properly coloured copies and rainbow copies of large graphs with small maximum degree
Let G be a graph on n vertices with maximum degree D. We use the Lov\'asz
local lemma to show the following two results about colourings c of the edges
of the complete graph K_n. If for each vertex v of K_n the colouring c assigns
each colour to at most (n-2)/22.4D^2 edges emanating from v, then there is a
copy of G in K_n which is properly edge-coloured by c. This improves on a
result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4),
409-433, 2003]. On the other hand, if c assigns each colour to at most n/51D^2
edges of K_n, then there is a copy of G in K_n such that each edge of G
receives a different colour from c. This proves a conjecture of Frieze and
Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a
framework developed by Lu and Sz\'ekely [Electron. J. Comb. 14(1), R63, 2007]
for applying the local lemma to random injections. In order to improve the
constants in our results we use a version of the local lemma due to Bissacot,
Fern\'andez, Procacci, and Scoppola [preprint, arXiv:0910.1824].Comment: 9 page
On small Mixed Pattern Ramsey numbers
We call the minimum order of any complete graph so that for any coloring of
the edges by colors it is impossible to avoid a monochromatic or rainbow
triangle, a Mixed Ramsey number. For any graph with edges colored from the
above set of colors, if we consider the condition of excluding in the
above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted
. We determine this function in terms of for all colored -cycles
and all colored -cliques. We also find bounds for when is a
monochromatic odd cycles, or a star for sufficiently large . We state
several open questions.Comment: 16 page
Ramsey properties of randomly perturbed graphs: cliques and cycles
Given graphs , a graph is -Ramsey if for every
colouring of the edges of with red and blue, there is a red copy of
or a blue copy of . In this paper we investigate Ramsey questions in the
setting of randomly perturbed graphs: this is a random graph model introduced
by Bohman, Frieze and Martin in which one starts with a dense graph and then
adds a given number of random edges to it. The study of Ramsey properties of
randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in
2006; they determined how many random edges must be added to a dense graph to
ensure the resulting graph is with high probability -Ramsey (for
). They also raised the question of generalising this result to pairs
of graphs other than . We make significant progress on this
question, giving a precise solution in the case when and
where . Although we again show that one requires polynomially fewer
edges than in the purely random graph, our result shows that the problem in
this case is quite different to the -Ramsey question. Moreover, we
give bounds for the corresponding -Ramsey question; together with a
construction of Powierski this resolves the -Ramsey problem.
We also give a precise solution to the analogous question in the case when
both and are cycles. Additionally we consider the
corresponding multicolour problem. Our final result gives another
generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we
determine how many random edges must be added to a dense graph to ensure the
resulting graph is with high probability -Ramsey (for odd
and ).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil
Powierski, stated results for cliques in graphs of low positive density
separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to
appear in CP
Ramsey numbers of Berge-hypergraphs and related structures
For a graph , a hypergraph is called a Berge-,
denoted by , if there exists a bijection such
that for every , . Let the Ramsey number
be the smallest integer such that for any -edge-coloring of
a complete -uniform hypergraph on vertices, there is a monochromatic
Berge- subhypergraph. In this paper, we show that the 2-color Ramsey number
of Berge cliques is linear. In particular, we show that for and where is a Berge-
hypergraph. For higher uniformity, we show that for
and for and sufficiently large. We
also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs
and expansion hypergraphs.Comment: Updated to include suggestions of the refere
Graph removal lemmas
The graph removal lemma states that any graph on n vertices with o(n^{v(H)})
copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite
its innocent appearance, this lemma and its extensions have several important
consequences in number theory, discrete geometry, graph theory and computer
science. In this survey we discuss these lemmas, focusing in particular on
recent improvements to their quantitative aspects.Comment: 35 page
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