Let denote the diagonal Ramsey number.

**Conjecture**exists.

**Problem**Determine the limit in the above conjecture (assuming it exists).

Erdos offered $100 for a solution to the highlighted conjecture and $250 for a solution to the associated problem (these prizes are now provided by Graham).

Classic results of Erdos [E] and Erdos-Szekeres [ESz] give bounds on which show that if exists, then it is in the interval . Although these arguments are quite basic, little progress has been made in improving these bounds. The best known lower bound on is due to Spencer [S] and the best known upper bound is due to Thomason [T]. They are as follows:

Gowers [G] has suggested that resolving these problems might require a rough structure theorem.

## Bibliography

[E] P. Erdos, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294. MathSciNet

[ESz] P. Erdos and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470.

[G] W. T. Gowers, Rough structure and classification, GAFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, Part I, 79--117. MathSciNet

[S] J. Spencer, Ramsey’s theorem—a new lower bound, J. Comb. Theory Ser. A 18 (1975), 108–115. MathSciNet

[T] A. Thomason, An upper bound for some Ramsey numbers, J. Graph Theory 12 (1988), 509–517. MathSciNet

* indicates original appearance(s) of problem.