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Statement

Theorem 2.4 (p. 8).

$$\log_2\log_2r_3(k,k)\le(2+o(1))k.$$

Here $r_3(k,k)$ is the least $N$ such that every red-blue coloring of the triples of an $N$-element set has a set of $k$ elements all of whose triples have the same color (p. 2), and $o(1)$ is as $k\to\infty$. The paper presents it as improving the bound $r_3(k,k)\le2^{2^{4k}}$ it attributes to Erdős and Rado (p. 8).

Source. D. Conlon, J. Fox and B. Sudakov, Hypergraph Ramsey numbers, arXiv:0808.3760v1, Theorem 2.4 on p. 8 (J. Amer. Math. Soc. 23 (2010), 247--266, not compared). The edition read is identified on the source card.

Read depth. Claims checked: the statement was read on the page image. The paper writes no proof, and none is checked or reconstructed here.

Proof pointer

None written. The paper says the theorem follows easily by taking $\alpha=1/2$ in Theorem 2.1 (p. 6), which bounds $r_3(s,n)$ by $(v+1)\alpha^{-r}(1-\alpha)^{r-m}$ in terms of the vertices $v$, red edges $r$ and total edges $m$ a builder needs in the vertex on-line Ramsey game; with $\alpha=1/2$ the bound is $(v+1)2^m$, free of $r$, and Lemma 2.2 (p. 7) bounds the edges the builder needs.

Dependencies

Theorem 2.1 and Lemma 2.2 of the same paper.

Bears on

• Problem 564: the problem asks for a lower bound $R_3(n)\ge2^{2^{cn}}$ for the same number $R_3(n)=r_3(n,n)$. The theorem is an upper bound of that doubly exponential shape, with top exponent $(2+o(1))n$; it says nothing about the lower bound the problem asks for.