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Statement

Setting (p. 552). For $d>3$ and $k\in\{0,1,\ldots,5\}$, $A_{d,k}$ is the configuration (5) of the $d$ real lines $L_{d,k,\nu}$, $\nu\in S$, where $L_{d,k,\nu}$ is the line $z=e^{-2\pi iu}+t\,e^{i\pi u}$, $t\in\mathbb R$, $u=(3\nu-k-1)/(3d)$, in the $(x,y)$ plane identified with $\mathbb C$, and $S=\{-m+1,\ldots,m+1\}$ if $d=2m+1$, $S=\{-m+1,\ldots,m\}$ if $d=2m$. The full construction is on the Theorem 1 page.

Lemma 1 (p. 552, quoted).

"1. Each line in $A_{d,k}$ intersects each other line.

2. $L_{d,k,\nu_1}\cap L_{d,k,\nu_2}\cap L_{d,k,\nu_3}\neq\emptyset$ iff $\nu_1+\nu_2+\nu_3\equiv k+1\ (\mathrm{mod}\ d)$, $\forall L_{d,k,\nu_1},L_{d,k,\nu_2},L_{d,k,\nu_3}\in A_{d,k}$.

3. There is no vertex of multiplicity higher than 3 in $A_{d,k}$."

Part 2 concerns three distinct lines of $A_{d,k}$, as its proof shows.

Proof pointer

P. 552. Part 1: the paper's condition for two lines to meet is that $(\nu_1-\nu_2)/d$ is not an integer, which holds because indices in $S$ differ by at most $d-1$. Part 2: the paper states the concurrency criterion $\cos\pi(2u_2+u_1)=\cos\pi(2u_3+u_1)$ for the parametrization (4) and rewrites it as $u_1+u_2+u_3\in\mathbb Z$, that is $\nu_1+\nu_2+\nu_3\equiv k+1\pmod d$. Part 3: a fourth line through a triple point would make two of the $u_i$ differ by an integer, which part 1 excludes.

Read depth

Claims checked: the statement was read clause by clause on the page image of the print and the proof was followed; the concurrency criterion in part 2 is stated in the paper without derivation and was not checked. Nothing here is independently reviewed.

Dependencies

None in the corpus. The parametrization (4) of the lines rests on the author's earlier papers.

Source. J. García Escudero, 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 edition read is named on the source card.

Bears on

• Problem 209: parts 1 and 3 give the arrangements $A_{d,k}$ the problem's hypotheses, every two lines meeting and no point on four or more lines; Theorem 1 supplies the absence of Gallai triangles.