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Does there exist a graph $G$ with at most $10^{10}$ many vertices which contains no $K_4$, and yet any $2$-colouring of the edges produces a monochromatic $K_3$?
Erdős and Hajnal [ErHa67] first asked for the existence of any such graph. Existence was proved by Folkman [Fo70], but with very poor quantitative bounds. (As a result these quantities are often called Folkman numbers.) Let this particular Folkman number be denoted by $N$.

Frankl and Rödl [FrRo86] proved $N\leq 7\times 10^{11}$, which Spencer [Sp88] improved to $\leq 3\times 10^{9}$. This resolved the initial challenge of Erdős [Er75d] to beat $10^{10}$.

Lu [Lu07] proved $N\leq 9697$ vertices. The current record is due to Dudek and Rödl [DuRo08] who proved $N\leq 941$ vertices. For further information we refer to a paper of Radziszowski and Xu [RaXu07], who prove that $N\geq 19$ and speculate that $N\leq 127$.

See also the entry in the graphs problem collection.