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Statement

Setting (pp. 85--86). A system $\Sigma_0: X_\mu\ (\mu\in M)$ is an indexed family of sets, not necessarily distinct. It is an $(a,b)$-system when $|M|=a$ and $|X_\mu|=b$ for every $\mu\in M$; the forms $(>a,\le b)$-system and so on are read in the obvious way. It is a $\Delta(a)$-system with kernel $K$ when $|M|=a$ and $X_{\mu_0}\cap X_{\mu_1}=K$ for all distinct indices $\mu_0,\mu_1\in M$ (for $|M|=1$ the kernel is required to lie inside the one set, and the empty system is a $\Delta(0)$-system with any kernel). Throughout, $a$ and $b$ are arbitrary cardinals, finite or infinite, and $b^+$ is the next larger cardinal after $b$.

Theorem I (p. 86).

• (i) If $a,b\ge1$, then every $(>b^+b^ba^{b+1},\le b)$-system contains a $\Delta(>a)$-system.
• (ii) If $a\ge2$, $b\ge1$ and $a+b\ge\aleph_0$, then every $(>a^b,\le b)$-system contains a $\Delta(>a)$-system.

The product $b^+b^ba^{b+1}$ in (i) is a product of cardinals; for finite $b$ the factor $b^+$ is $b+1$.

By Remark 1 (p. 86), Theorem II shows that (ii) is best possible: for $a\ge2$, $b\ge1$, $a+b\ge\aleph_0$ not every $(a^b,\le b)$-system contains a $\Delta(>a)$-system.

Proof pointer

Pp. 87--89. The proof of (i) supposes a $(|N|,\le b)$-system with no $\Delta(>a)$-system and shows $|N|\le b^+b^ba^{b+1}$, inequality (3) on p. 87. For each index it builds, by transfinite recursion over the ordinals of cardinality at most $b$, a sequence of elements of the sets, each step working inside a maximal $\Delta$-subfamily with the kernel built so far, which has at most $a$ members by hypothesis. The Ramification Lemma (p. 86) then bounds the number of index classes: there are at most $(ba)^{|\alpha_0|}$ distinct vectors for each length $\alpha_0$, giving $|N|\le a(ba)^bb^+$ (p. 89). Part (ii) follows from (i) because $b^+b^ba^{b+1}=a^b$ when $a\ge2$, $b\ge1$ and $a+b\ge\aleph_0$ (p. 89).

Read depth

Claims checked: the definitions on p. 85, Theorem I and Remark 1 were read clause by clause on the page images of the print, and the outline of the proof on pp. 87--89 was followed. Nothing here is independently reviewed.

Dependencies

The paper's Ramification Lemma (p. 86), stated on the source card. Optimality of (ii) comes from Theorem II.

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: for finite $a$ and $b$, part (i) gives a threshold $(b+1)b^ba^{b+1}$, weaker than the bound of Theorem III, which is the paper's statement that concerns the problem.