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Statement

The terms system, $(>c,\le b)$-system and $\Delta(>a)$-system are those of Theorem I (p. 85): a system is an indexed family whose sets need not be distinct, and a $\Delta(>a)$-system is a subsystem of more than $a$ members whose pairwise intersections, over distinct indices, all equal one set.

Theorem III (p. 86). Let $a$ and $b$ be integers with $1\le a,b<\aleph_0$, and put

$$c=b!\,a^{b+1}\Bigl(1-\frac{1}{2!\,a}-\frac{2}{3!\,a^2}-\cdots-\frac{b-1}{b!\,a^{b-1}}\Bigr) \qquad(1)$$

Then every $(>c,\le b)$-system contains a $\Delta(>a)$-system.

Remarks on p. 86:

• For $a=b=2$ the result is best possible: here $c=12$, and the paper gives a $(12,2)$-system with no $\Delta(3)$-system, made of six pairs each listed twice.
• For $a=3$, $b=2$ the paper says Theorem III is not best possible.
• By Theorem II, Theorem III is best possible except for a factor between $1$ and $b!$.

The paper's conjecture that $b!$ in (1) can be replaced by $c_1^b$ has its own page, Conjecture (p. 86).

Proof pointer

Pp. 89--90. Let $f(a,b)$ be the least threshold, finite by Theorem I, and $\phi(a,b)$ the least number such that every $(>\phi,\le b)$-system of pairwise distinct sets contains a $\Delta(>a)$-system. Since copies of one set form a $\Delta$-system, each set occurs at most $a$ times, giving $f(a,b)\le a\,\phi(a,b)$, inequality (6). For distinct sets, a maximal pairwise disjoint subfamily has at most $a$ members; every other set meets their union, and removing a common point $\xi$ reduces to sets of at most $b-1$ elements. This gives $\phi(a,b)\le a+(\phi(a,b-1)-1)\,ba$, which with $\phi(a,1)=a$ and $b-1$ iterations yields $\phi(a,b)\le c/a$, and so $f(a,b)\le c$.

Read depth

Claims checked: Theorem III, formula (1), the remarks on p. 86 and the proof on pp. 89--90 were read clause by clause on the page images of the print. The arithmetic $c=12$ for $a=b=2$ was rechecked here. Nothing here is independently reviewed.

Dependencies

Theorem I, for the finiteness of the threshold.

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 $a=k-1\ge1$, a family of more than $c$ distinct $n$-element sets contains $k$ sets forming a sunflower, so $f(n,k)\le c+1$. For $k\ge3$ this bound is of order $n!\,(k-1)^{n+1}$, not of the form $c_k^n$ the problem asks for; for $k=2$ formula (1) gives $c=1$. For families of distinct sets the proof on p. 90 gives the smaller threshold $\phi(k-1,n)\le c/(k-1)$; the paper does not state this as a theorem.