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Statement

Section 4, p. 106, at the close of the survey of ordinary lines. Erdős writes: "Does there exist for every $k$ a set of points in the plane so that if one colors the points by two colors in an arbitrary way, there always should be at least one line which contains at least $k$ points and all whose points have the same color." He reports that Graham and Selfridge gave an affirmative answer for $k=3$, and that the cases $k>3$ seem to be open.

The question as printed does not say that the set is finite; the points counted on the line are, in the natural reading, points of the set. The report on $k=3$ gives neither the construction nor a reference.

Source. P. Erdős, On some problems of elementary and combinatorial geometry, Ann. Mat. Pura Appl. (4) 103 (1975), 99-108; Section 4, p. 106. The edition read is identified on the source card.

Read depth. Claims checked: the question and the report were read clause by clause on the page image of p. 106.

Proof pointer

None; the survey states the question and the report only.

Dependencies

None.

Bears on

• Problem 1090: the question is the problem's, which the site states for finite sets and $k\ge3$. The survey's report is the Graham-Selfridge case $k=3$, without construction or reference, and it records the cases $k>3$ as open in 1975.