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Statement

Theorem 6.2 (p. 16). For $\eta>0$ and all positive integers $r$ and $k$ there is a constant $\beta=\beta(r,k,\eta)>0$ such that every coloring of the $k$-element subsets of an $N$-element set with $r$ colors has a subset of size $s>(\log N)^\beta$ more than $(1-\eta)\binom sk$ of whose $k$-element subsets have one color.

The paper sets it against a remark of Erdős (p. 16): he would begin to doubt that $r_3(n,n)$ is doubly exponential in $n$ if every two-coloring of the triples of an $N$-set had a set of size $s=c(\eta)(\log N)^\epsilon$ with at least $(1-\eta)\binom s3$ triples of one color. The paper says the theorem gives this when $\epsilon$ is allowed to decrease with $\eta$; here $\beta$ depends on $\eta$.

Consequence stated in the paper (p. 17). For each $\alpha>0$ and $k$ there are $c,\epsilon>0$ with $F^{(k)}(N,\alpha)>c(\log N)^\epsilon$, where $F^{(k)}(N,\alpha)$ is the threshold function defined on p. 16 and recorded, with the paper's wording of its definition, on the Section 6.2 page. The paper states it as what Theorem 6.2 "demonstrates" for $\alpha$ bounded away from $0$ and writes out no further argument.

Source. D. Conlon, J. Fox and B. Sudakov, Hypergraph Ramsey numbers, arXiv:0808.3760v1, Section 6.2: Theorem 6.2 on p. 16, the consequence and Theorem 6.3 on p. 17, the proof of Theorem 6.3 on pp. 17--18 (J. Amer. Math. Soc. 23 (2010), 247--266, not compared). The edition read is identified on the source card.

Read depth. Claims checked: the statements of Theorems 6.2 and 6.3 and the consequence were read clause by clause on the page images. The proof of Theorem 6.3 was read for the outline below; its steps were not checked. The paper deduces Theorem 6.2 from it only through the remark on edge density below.

Proof pointer

The paper deduces Theorem 6.2 from Theorem 6.3 (p. 17): for all positive integers $r,k,\ell$ there is $c=c(r,k,\ell)$ with $r(K_\ell^{(k)}(n);r)\le e^{cn^\ell}$, where $K_\ell^{(k)}(n)$ is the $k$-uniform hypergraph on $\ell$ parts of size $n$ whose edges are the $k$-sets meeting $k$ different parts, and $r(H;r)$ is the least $N$ such that every $r$-coloring of the $k$-sets of an $N$-set has a monochromatic copy of $H$. The blow-up has $\ell n$ vertices and at least $\left(1-\binom k2/\ell\right)\binom{\ell n}k$ edges (p. 17), so its density tends to $1$ as $\ell$ grows; the paper calls Theorem 6.2 a corollary. The exponent of Theorem 6.3 is printed as $cn^\ell$; the proof's first line takes $N=e^{cn^{\ell-1}}$. The proof of Theorem 6.3 (pp. 17--18) counts the monochromatic $\ell$-sets forced by the $r$-color Ramsey number of $K_\ell^{(k)}$, takes a popular color, and applies an extremal lemma for dense $\ell$-uniform hypergraphs (cited to Erdős and to Nikiforov, the paper's [8] and [25]) to find a complete $\ell$-partite $\ell$-uniform hypergraph with parts of size $n$ in that color.

Dependencies

Theorem 6.3 of the same paper; the counting trick the paper credits to its references [10] and [21]; the extremal lemma of its references [8] and [25].

Bears on

• Problem 161: the problem asks whether, for fixed $k$, the growth of $F^{(k)}(N,\alpha)$ changes continuously as $\alpha$ runs from $0$ to $1/2$ or jumps. The consequence above gives a lower bound of a power of $\log N$ for every fixed $\alpha>0$ and every $k$. It does not decide whether a jump occurs, and the paper states the bound, not an answer.
• Problem 564: the paper relates the theorem to Erdős's remark above on the growth of $r_3(n,n)$; the theorem gives no bound on $r_3(n,n)$.