---

Source. Theorem 1, pp. 128-129, 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.

Setting

$X$ is a set of $n$ elements. A family $\mathcal F$ of $k$-subsets of $X$ is an $(n,L,k)$-system if any two different members meet in a number of elements belonging to $L$ (p. 125), and $\mathcal F_x=\{F-x : x\in F\in\mathcal F\}$. The paper introduces Theorem 1 (p. 128) as a slightly weaker result which implies the conjecture of Erdős and Sós.

Statement

Theorem 1 (pp. 128-129). Let $\mathcal F$ be an $(n,\{0,2,3,\ldots,k-1\},k)$-system of subsets of $X$, with $n>n_0(k)$. Then one of the following cases occurs:

(i) $\lvert\mathcal F\rvert<\binom{n-2}{k-2}$;

(ii) for some $x\ne y$ in $X$, $\mathcal F$ is the family of all $k$-subsets of $X$ containing both $x$ and $y$;

(iii) for some $x\in X$, $\lvert\mathcal F_x\rvert<\binom{n-3}{k-3}$;

(iv) for some $x\ne y$ in $X$, fewer than $\binom{n-3}{k-3}+\binom{n-4}{k-3}$ members of $\mathcal F$ meet $\{x,y\}$.

The printed hypothesis does not repeat $k\ge4$, and $n_0(k)$ is left implicit. Lemmas 1 and 2 (pp. 126-127), on which the proof rests, assume $k\ge4$ (Lemma 3, pp. 127-128, does not), and the proof uses $k\ge4$ on p. 129.

Read depth. Claims checked: the statement was read clause by clause on the print and the proof read through.

Proof pointer

Pages 129-132, by contradiction. Assuming (iii) and (iv) fail, the paper uses Lemmas 1 to 3 on the minimal sunflower kernels of the links to show that each link's kernels form a single pair or a single point. Then $\mathcal F$ lies in a family built from disjoint pairs $\{x_i,y_i\}$ and classes $Z_i$, and a count by induction on the number of pairs gives (i) or (ii).

Bears on

• Problem 702: the structural step from which Theorem 2 derives the problem's corrected Statement; on its own it does not give the bound.