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Statement

The paper's notation and the set $S(k,l,m)$ of systems are those of Theorem 1; $S(2,2r,4r)$ consists of the systems of subsets of $[0,4r)$, each of at most $2r$ elements, no member containing another, any two members sharing at least two elements.

Example (p. 319). For $r>0$, the $2r$-subsets $a$ of $[0,4r)$ with $\lvert a\cap[0,2r)\rvert>r$ form a system in $S(2,2r,4r)$ with

$$n=\frac12\binom{4r}{2r}-\frac12\binom{2r}{r}^2$$

members. The paper notes that this exceeds $\binom{4r-2}{2r-2}$, the bound of Theorem 2 (b), for every large $r$, possibly for every $r>2$. The paper introduces it as a more general example than S. H. Min's example of p. 318: the 4-subsets $a$ of $[0,8)$ with $\lvert a\cap[0,4)\rvert=3$, sixteen sets forming a system in $S(2,4,8)$, against $\binom62=15$.

Conjecture (p. 319). The authors conjecture that for these values of $k,l,m$ the example is a case of largest $n$: if $r>0$ and $(a_0,\ldots,a_{n-1})\in S(2,2r,4r)$, then

$$n\le\frac12\binom{4r}{2r}-\frac12\binom{2r}{r}^2.$$

The conjecture covers systems whose members may have fewer than $2r$ elements, provided no member contains another; the example has all members of size $2r$.

Source. P. Erdős, Chao Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2) 12 (1961), 313–320, as identified on the source card: concluding remark (i), pp. 318–319, with the conjecture on p. 319.

Read depth. Claims checked: the example, its count and the conjecture were read clause by clause on the print. Nothing here is independently reviewed.

Proof pointer

A conjecture; the paper proves only the count of the example, by summing $\binom{2r}\lambda\binom{2r}{2r-\lambda}$ over $r<\lambda\le2r$ and using the symmetry $\lambda\leftrightarrow2r-\lambda$ (p. 319).

Dependencies

None.

Bears on

• Problem 83: the problem's statement, with its $n$ the paper's $r$, is the conjecture restricted to systems all of whose members have exactly $2r$ elements, for which the incomparability condition is automatic. The paper poses the conjecture and gives the example showing the bound would be attained. For $r\ge2$ it proves no bound of that size: Theorem 2 (a) applies to this case, but its bound is larger (that comparison is arithmetic, not stated in the paper).