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Escudero answers the question of Problem 209 in the negative for every $d\ge 4$: there is an arrangement of $d$ real lines, no two parallel, with no point on four or more of the lines and with no Gallai triangle, a triangle formed by three of the lines whose three corners each lie on exactly two lines of the arrangement. Theorem 1 of the paper exhibits explicit arrangements $A_{d,k}$ with no Gallai triangle for $d=3q+1$ and $d=3q+2$ with $k=0$, for $d=9n$ with $k=0$, and for $d=3q$ with $q$ not divisible by $3$ and $k=2$; together these cover every $d\ge 4$. The lines come from the folding polynomials of the affine Weyl group of the root lattice $A_2$, and the absence of a Gallai triangle reduces to the unsolvability of linear congruences modulo $d$. The source card [[../library/discrete_geometry/escudero_2016_gallai_triangles_configurations_lines_projective_plane/_index|records the paper's statements]]. The earlier arrangements of Füredi and Palásti covered every $d\ge 4$ not divisible by $9$; this result adds the multiples of $9$.

The result is refereed: Juan García Escudero, Gallai triangles in configurations of lines in the projective plane, C. R. Math. Acad. Sci. Paris 354 (2016), no. 6, 551-554. The site's curator, T. F. Bloom, marks the problem disproved and credits this paper with the answer for every $d\ge 4$. The site's label carries a Lean qualification: the formal_proof attribute of the formal-conjectures file, points to a Lean development that declares itself a formalization of Escudero's construction (the formalization link above, at its pinned commit). This corpus has not built or audited that development, so it is recorded as a link and not as evidence. The corpus records no check of the proof.