A new upper bound for diagonal Ramsey numbers

@article{Conlon2006ANU,
  title={A new upper bound for diagonal Ramsey numbers},
  author={David Conlon},
  journal={Annals of Mathematics},
  year={2006},
  volume={170},
  pages={941-960}
}
  • D. Conlon
  • Published 30 July 2006
  • Mathematics
  • Annals of Mathematics
We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant C such that r(k + 1,k+1) ≤ k -Clog k/log log k ( 2k k ). 

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References

SHOWING 1-9 OF 9 REFERENCES

An upper bound for some ramsey numbers

New upper bounds for the ramsey numbers r(k, l) are obtained. In particular it is shown there is a constant A such that

Quasi-random graphs

A large equivalence class of graph properties is introduced, all of which are shared by so-called random graphs, and it is often relatively easy to verify that a particular family of graphs possesses some property in this class.

Pseudo-Random Graphs

On a Problem of Formal Logic

This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given

Random Graphs

Numbers in ramsey theory, Surveys in Combinatorics, London Math. Soc. Lecture Note Series no

  • Numbers in ramsey theory, Surveys in Combinatorics, London Math. Soc. Lecture Note Series no
  • 1987

Pseudorandom graphs, Random Graphs '85 (Pozna´Pozna´n

  • Pseudorandom graphs, Random Graphs '85 (Pozna´Pozna´n
  • 1985

RÖDL, Numbers in Ramsey theory, in Surveys in Combinatorics

  • (C. WHITEHEAD, ed.), London Math. Soc. Lecture Note Ser
  • 1987

SZEKERES, A combinatorial problem in geometry

  • Compositio Math
  • 1935