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Source. Unnumbered opening statement, p. 125, of P. Frankl, "On families of finite sets no two of which intersect in a singleton," Bull. Austral. Math. Soc. 17 (1977), no. 1, 125-134, doi:10.1017/S0004972700025521. Pages are the journal's own, as on the source card.

Statement

Let $X$ be a set of $n$ elements and $\mathcal F$ a family of $k$-element subsets of $X$. The paper proves (p. 125): if $n>n_0(k)$, $k\ge4$ and $\lvert\mathcal F\rvert>\binom{n-2}{k-2}$, then $\mathcal F$ has two members $F,G$ with $\lvert F\cap G\rvert=1$.

Equivalently, in the language of §1 (pp. 125-126): an $(n,\{0,2,3,\ldots,k-1\},k)$-system, a family of $k$-subsets of an $n$-set in which any two different members meet in a number of elements from $\{0,2,3,\ldots,k-1\}$, has at most $\binom{n-2}{k-2}$ members when $k\ge4$ and $n\ge n_0(k)$, as the conjecture is restated there. The paper attributes the conjecture to Erdős and Sós, citing Erdős's problem paper in the Proceedings of the Fifth British Combinatorial Conference (1975), and records that Katona proved the case $k=4$ (unpublished). The bound is attained by the $\binom{n-2}{k-2}$ $k$-sets containing two fixed elements, any two of which share at least two elements.

The paper gives no explicit value of $n_0(k)$. The conclusion is delivered by Theorem 2 (p. 132), whose range is $n>n_0(k)+2\binom{n_0(k)}{k}$ with $n_0(k)$ the bound from Theorem 1.

Read depth. Claims checked: the statement and the definitions it uses were read on the print.

Proof pointer

Lemmas 1 to 3 (pp. 126-128) and Theorems 1 and 2 (pp. 128-133). The condition says exactly that each link $\mathcal F_x=\{F-x : x\in F\in\mathcal F\}$ is an intersecting family. The proof studies the minimal kernels of large sunflowers ($\Delta$-systems) in each link, proves the structural Theorem 1, and iterates it in Theorem 2.

Bears on

• Problem 702: this statement, with its range $n>n_0(k)$, is the problem's corrected Statement, and the paper proves it. The paper says nothing about smaller $n$.