Two extensions of Ramsey's theorem

@article{Conlon2011TwoEO,
  title={Two extensions of Ramsey's theorem},
  author={David Conlon and Jacob Fox and Benny Sudakov},
  journal={ArXiv},
  year={2011},
  volume={abs/1112.1548}
}
Ramsey's theorem, in the version of Erd\H{o}s and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,...,n} contains a monochromatic clique of order 1/2\log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of R\"odl, we show that there is a constant $c>0$ such that every 2-coloring of the edges of the complete graph on \{2, 3,...,n\} contains a monochromatic clique S for which the sum of 1/\log i over all vertices… 
2 Citations
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References

SHOWING 1-10 OF 27 REFERENCES
On two problems in graph Ramsey theory
TLDR
This work improves the upper bound on the existence of a constant c such that, for any graph H on n vertices, rind(H) ≤ 2cnlogn, and moves a step closer to proving this conjecture.
On graphs with small Ramsey numbers
Let R(G) denote the minimum integer N such that for every bicoloring of the edges of KN, at least one of the monochromatic subgraphs contains G as a subgraph. We show that for every positive integer
On homogeneous sets of positive integers
  • V. Rödl
  • Mathematics
    J. Comb. Theory, Ser. A
  • 2003
A few remarks on Ramsey-Tura'n-type problems
Hypergraph Ramsey numbers
The Ramsey number rk(s, n) is the minimum N such that every red-blue coloring of the k-tuples of an N -element set contains a red set of size s or a blue set of size n, where a set is called red
On graphs with small Ramsey numbers *
TLDR
It is shown that for every positive integer d and each,0< <1, there exists kˆ k (d, ) such that forevery bipartite graph Gˆ (W,U;E ) with the maximum degree of vertices in W at most d and jU j j jW j, R (G ) k jWJ.
Combinatorial Theorems on Classifications of Subsets of a Given Set
Given any positive integers k, n, ANT, there is a positive integer M which has the following property. If S = {1, 2, . . ., A1}, and A is any distribution of Q,(S) into k classes, there is always an
Partition relations for cardinal numbers
1. INTRODUCTION In this paper our main object is the study of relations between cardinal numbers which are written in the form a-(b o , b,,. . .) r or a-(b)C ~° or (b) (b o b,) Such relations were
Some remarks on the theory of graphs
The present note consists of some remarks on graphs. A graph G is a set of points some of which are connected by edges. We assume here that no two points are connected by more than one edge. The
...
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