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Statement

Conjecture (p. 86, unnumbered, quoted). "It is not improbable that in (1) the factor $b!$ can be replaced by $c_1{}^b$, for some absolute positive constant $c_1$."

Here (1) is the threshold $c=b!\,a^{b+1}\bigl(1-\frac{1}{2!\,a}-\cdots-\frac{b-1}{b!\,a^{b-1}}\bigr)$ of Theorem III, for finite $a,b\ge1$. The paper adds (p. 86) that such a sharpened form of Theorem III would have applications in number theory, and that those applications first led to the investigation.

Read literally, the sharpened statement says that every $(>c_1^b\,a^{b+1}(1-\cdots),\le b)$-system contains a $\Delta(>a)$-system, with $c_1$ independent of both $a$ and $b$.

Read depth

Claims checked: the sentence and its context on p. 86 were read on the page image of the print. The paper offers no argument for it.

Dependencies

Theorem III, whose formula (1) it modifies.

Source. P. Erdős and R. Rado, Intersection theorems for systems of sets, J. London Math. Soc. 35 (1960), 85--90, doi:10.1112/jlms/s1-35.1.85; the edition read is named on the source card.

Bears on

• Problem 20: with $b=n$ and $a=k-1$, the conjecture would give $f(n,k)\le c_1^n(k-1)^{n+1}+1$, a bound of the form $c_k^n$, so it implies a positive answer to the problem; the problem asks only for some $c_k$ depending on $k$, which is weaker than the conjecture's constant $c_1$ independent of $a$.