Escudero 2016 gallai triangles configurations lines projective plane

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lemma_1: García Escudero's lemma that in his arrangement A_{d,k} every two lines intersect, three lines are concurrent exactly when their indices sum to k+1 modulo d, and no vertex lies on more than three lines.

theorem_1: García Escudero's theorem that his real line arrangements A_{d,k} have no Gallai triangle for d = 3q+1, 3q+2 or 9n with k = 0 and for d = 3q with q not a multiple of 3 and k = 2, which with Lemma 1 answers Erdős's Gallai triangle question negatively for every d >= 4.

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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, DOI 10.1016/j.crma.2016.03.003. The copy read for this card prints "© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.", every other right reserved.

Erdős asked whether a line arrangement A in the projective plane in which no vertex lies on more than three lines of A must contain a Gallai triangle, that is, a triangle formed by three lines of A whose three intersection points all have multiplicity two (p. 551, Definition 1 and the Erdős question). Füredi and Palásti's arrangements B_n had already answered this negatively for every n >= 4 not divisible by nine (p. 552). The paper's arrangements come from the real one-parameter family of polynomials J_{d,tau} built from the A_2 folding polynomials: by the author's earlier work, cited rather than proved here, J_{d,tau} is a union of d lines located through the critical points of an associated trigonometric function H_{d,tau}, and A_{d,k} is the configuration of those lines at tau = (2k+1)pi/6, given parametrically by (4) and indexed by the set S of (5) (p. 552). Lemma 1 (p. 552) shows that every two lines of A_{d,k} meet, that three are concurrent exactly when their indices sum to k+1 modulo d, and that no vertex has multiplicity above three; Lemma 2 (p. 552) characterizes the vertices of multiplicity two by a linear congruence. Theorem 1 (p. 553) shows that A_{d,k} has no Gallai triangle for d = 3q+1 and d = 3q+2 (q >= 1) with k = 0, for d = 9n (n >= 1) with k = 0, and for d = 3q with q != 3n (3 does not divide q) and k = 2; the proof (p. 554) reduces to the solvability of congruences such as 9 nu_0 = 3 (mod d) for k = 0, using Propositions 1 and 2 (p. 553), standard facts on linear congruences that the paper cites. The closing remark (p. 554) concludes that for each integer d >= 4 there are configurations of d lines in the plane with no more than three lines through each vertex and no Gallai triangle, which answers Erdős's question; the abstract (p. 551) states the answer is negative for all d > 3.

Source: https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.03.003/.

Read status: claims checked for Definition 1, the construction (4)--(5), Lemmas 1 and 2, Theorem 1 and the closing remark, read clause by clause on the page images of the print; the proofs of Lemma 1 and Theorem 1 followed. The derivation of the lines from J_{d,tau} rests on the author's earlier papers and was not checked. Nothing here is independently reviewed. Result pages: theorem_1 and lemma_1.

Bears on. #209: Theorem 1 (p. 553) with Lemma 1 (p. 552) gives, for every d >= 4, an arrangement of d real lines, each meeting every other, with no point on four or more of them and no Gallai triangle; the paper states that this answers Erdős's question (p. 554), in the negative for every d > 3 (abstract, p. 551).

Results.

• Theorem 1 (p. 553): the configurations A_{d,k} have no Gallai triangles for (d = 3q+1, k = 0), (d = 3q+2, k = 0), (d = 9n, k = 0) and (d = 3q with q != 3n, k = 2), where q, n >= 1; with the closing remark (p. 554) that this covers every d >= 4.
• Lemma 1 (p. 552): in A_{d,k} every two lines meet, three lines are concurrent iff their indices sum to k+1 modulo d, and no vertex has multiplicity higher than 3.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.