# 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} }

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

- MathematicsJ. Graph Theory
- 1988

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

- Mathematics, Computer ScienceComb.
- 1989

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.

### On a Problem of Formal Logic

- Mathematics
- 1930

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…

### 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