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Source. The paragraphs on pp. 38--39 of P. Erdős, Combinatorial problems in geometry, Math. Chronicle 12 (1983), 35--54, the transcript of an invited address at the 17th New Zealand Mathematics Colloquium (Dunedin, 17--19 May 1982), as named on the source card. The lecture numbers none of its statements; the pages are the journal's own.

Statement

Setting (pp. 36--37). The lecture first states the Sylvester--Gallai theorem: if $n$ points in the plane are not all on a line, some line passes through exactly two of them. A line through exactly two of the points is an ordinary line (the paper's term, p. 38).

Reported results (pp. 38--39). For $n$ points in the plane, not all on a line:

1. De Bruijn and Erdős conjectured that the number of ordinary lines tends to infinity with $n$; the paper says Motzkin proved this in 1951 in the Transactions.
2. Kelly and Moser proved that there are at least $3n/7$ ordinary lines (the print writes "there are $3n/7$ lines which go through two"), and the paper says this is best possible at least for seven points, by the seven-point configuration drawn on p. 38, which has three ordinary lines.
3. Motzkin conjectured that for $n$ greater than thirteen there are at least $n/2$ ordinary lines, and showed that this is best possible for even $n$ if true. The paper reports that Hansen had very recently proved the conjecture in a proof of about fifty pages that Erdős had not seen, and that it seemed to have been checked carefully by Fenchel.

Read depth. Claims checked: the passage was read clause by clause on the page images of the print. A second reader checked the statement, hypotheses, label and page against the print.

Proof pointer

The paper proves none of these; it reports them. It gives Kelly's proof of the Sylvester--Gallai theorem (pp. 37--38) only.

Dependencies

None.

Bears on

• Problem 210: the problem asks whether the least number of ordinary lines tends to infinity and how fast. The paper reports Motzkin's affirmative answer, the Kelly--Moser bound $3n/7$ and the claim that Hansen proved the $n/2$ bound for $n>13$; it proves none of them, and it is not a source for them.