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Füredi and Palásti construct, for every $d\ge 4$ that is not divisible by $9$, an arrangement of $d$ real lines, no two parallel, in which no point lies on four or more of the lines and no three of the lines form a Gallai triangle, that is, a triangle whose three corners each lie on exactly two lines of the arrangement. Any one of these arrangements answers the question of Problem 209 in the negative, since the question asks whether every admissible arrangement must contain a Gallai triangle; the arrangements with $d=4$ lines already do so. The construction appears in Z. Füredi and I. Palásti, Arrangements of lines with a large number of triangles, Proc. Amer. Math. Soc. 92 (1984), no. 4, 561-566, whose arrangements were built to carry many triangles. The values $d$ divisible by $9$ were covered later by Escudero's arrangements.

The result is refereed, published in the Proceedings of the American Mathematical Society. The site's curator, T. F. Bloom, marks the problem disproved and credits this paper with the cases $d\ge 4$ not divisible by $9$. The corpus records no check of the proof.