Non-simple Arrangements of Lines and Pseudolines with Many Triangles

We study affine arrangements of an even number n of lines and pseudolines with many triangles. For simple ones, Blanc's bound limits the number of triangular faces to lfloor n(n-5/2)/3rfloor. Adapting the Bartholdi-Blanc-Loisel doubling to the even case, we construct an infinite series of arrangements of n=6cdot2^t+2 (8,14,26,50,ldots) straight lines with two triple points and (n(n-5/2)+1)/3 triangles, one more than the simple bound allows. We obtain exhaustive enumerations of triangle-maximal pseudoline arrangements, simple or not, for even nle12. For n=6,8,10,12,14 the maximum number of triangles, for both lines and pseudolines, is 7,15,25,38,54; for n=8,12,14 every optimal arrangement has multiple points. All optimal arrangements for even nle12 are realizable by straight lines, except two arrangements of 10 pseudolines, which are not stretchable. For the odd value n=11, no arrangement with 33 triangles exists even with multiple points. We also obtain even arrangements by adding a line to odd ones. For nequiv2pmod 6 this again gives one triangle more than the simple bound, in pseudolines up to n=44 and in straight lines for n=20 and n=32. Doubling these pseudoline arrangements gives members of the series 18cdot2^t+2 and 30cdot2^t+2 with the same excess, up to n=146 and n=242, respectively; those with n=38 and n=74 are realizable by straight lines. These results lead us to conjecture that an arrangement of an even number n of pseudolines has at most lfloor n(n-2)/3rfloor-lfloor n/6rfloor triangles.