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Statement

The paper restates the results of the section in terms of a function it says Erdős introduced in its reference [11], defined as printed on p. 16 (arXiv v1; page image): "Denote by $F^{(k)}(N,\alpha)$ the largest integer for which it is possible to split the $k$-tuples of a $N$-element set $S$ into two classes so that for every $X\subset S$ with $|X|\ge F^{(k)}(N,\alpha)$, each class contains more than $\alpha\binom{|X|}{k}$ $k$-tuples of $X$. Note that $F^{(k)}(N,0)$ is essentially the inverse function of the usual Ramsey function $r_k(n,n)$. It is easy to show that for $0\le\alpha<1/2$,

$$c(\alpha)\log N<F^{(2)}(N,\alpha)<c'(\alpha)\log N.$$

"

The paper's [11] is Erdős's 1990 chapter (problem_p21), which defines the function as the smallest integer for which such a split is possible, the wording the site's Problem 563 uses. An observation made here: the property "some split has every class dense on every $X$ with $|X|\ge m$" is preserved when $m$ increases, so the admissible thresholds $m$ form an up-set, the smallest of them is the meaningful quantity, and "largest" cannot be read as printed (every $m>N$ is admissible vacuously). This page records the paper's wording as printed and does not equate the two definitions silently; the bound displayed is the same as Erdős's display (29) for $k=2$ classes, and neither source proves it.

Source. D. Conlon, J. Fox and B. Sudakov, Hypergraph Ramsey numbers, arXiv:0808.3760v1, Section 6.2, p. 16 (J. Amer. Math. Soc. 23 (2010), 247--266, not compared); read on the page image of the preprint. No file of either version is held.

Read depth. Claims checked: the definition and the display were read clause by clause on the page image. No proof is given in the source ("It is easy to show"); none is reconstructed here.

Proof pointer

None in the source. The surrounding section deduces the hypergraph statement Theorem 6.2 from Theorem 6.3 (p. 17); that argument does not concern the graph bound above.

Dependencies

None stated.

Bears on

• Problem 563: the display is the known two-sided bound $F(n,\alpha)\asymp_\alpha\log n$ the site's commentary quotes; the asymptotic $F(n,\alpha)\sim c_\alpha\log n$ the problem asks for is not addressed in the paper.
• Problem 162: the site's wording prints "largest", as this paper does, with the range $0\le\alpha\le1/2$; corrected to "smallest" and $0\le\alpha<1/2$, its question is that of Problem 563. The display is the known two-sided bound, and the asymptotic the problem asks for is not addressed in the paper.