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Statement

Section 6, p. 108.

Triples. "V. T. Sós and I proved that if there are $n+1$ triples in a set $S$ of $n$ elements, then there are always two of them whose intersection is a singleton, for $n\equiv0\pmod 4$ this is best possible." The proof is left to the reader.

The conjecture. "We conjectured that if $l>3$, $A_i\subset S$, $1\leqslant i\leqslant k$, $|A_i|=l$, $n>n_0(l)$, $k>\binom{n-2}{l-2}$ then for some $1\leqslant i<j\leqslant k$, $|A_i\cap A_j|=1$." Here $S$ is the set of $n$ elements above and $n_0(l)$ a threshold depending on $l$. Erdős adds that the conjecture, if true, is best possible, as the $\binom{n-2}{l-2}$ sets of size $l$ containing two fixed elements of $S$ show: any two of them share at least those two elements.

Status in the survey. Erdős reports that Katona proved the conjecture for $l=4$, by an unpublished proof that is "not very simple", and that "The cases $l>4$ are open."

Source. P. Erdős, On some problems of elementary and combinatorial geometry, Ann. Mat. Pura Appl. (4) 103 (1975), 99-108; Section 6, p. 108. The edition read is identified on the source card.

Read depth. Claims checked: the triple theorem, the conjecture and the report on it were read clause by clause on the page image of p. 108.

Proof pointer

None in the survey: the triple case is left to the reader, the extremal example is the one described above, and Katona's proof for $l=4$ is reported as unpublished.

Dependencies

None.

Bears on

• Problem 702: the conjecture is the problem's statement with $k$ in place of $l$ and with Erdős's range $n>n_0(l)$, the range the problem's corrected Statement takes from Erdős's texts; the site's wording omits it. The survey reports the case $l=4$ as proved by Katona, unpublished, and the cases $l>4$ as open in 1975.