Problem 209

Let $A$ be a finite collection of $d\geq 4$ non-parallel lines in $\mathbb{R}^2$ such that there are no points where at least four lines from $A$ meet. Must there exist a 'Gallai triangle' (or 'ordinary triangle'): three lines from $A$ which intersect in three points, and each of these intersection points only intersects two lines from $A$ ?

Source: erdosproblems.com/209

Geometry

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SolvedDisproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN). The question as stated is refuted already by an elementary arrangement of four lines (Current assessment); the site credits Füredi and Palásti and Escudero, whose arrangements together cover every $d\ge4$ .

Er84: Erdős, P., Research problems. Period. Math. Hungar. 15 (1984), 101-103; the site's first source key for the problem.
ErPu95b: Erdős, P. and Purdy, G., Extremal problems in combinatorial geometry. Handbook of Combinatorics, Vol. 1, Elsevier (1995), 809-874, p. 818; the site's second source key, and the reference Escudero gives for the question.
Es16: Escudero, Juan García, Gallai triangles in configurations of lines in the projective plane. C. R. Math. Acad. Sci. Paris 354 (2016), no. 6, 551-554.
FuPa84: Füredi, Z. and Palásti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.

Problem 209

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