Roberts's triangle theorem

Branko Grünbaum found the proof in Roberts's original paper "unconvincing",^[3][4] and credits the first correct proof of Roberts's theorem to Robert W. Shannon, in 1979.^[1][5] He presents instead the following more elementary argument, first published in Russian by Alexei Belov.^[1][6] It depends implicitly on a weaker version of the same theorem, according to which every simple arrangement of three or more lines has at least one triangular face. This follows easily by induction from the fact that adding a line to an arrangement cannot decrease the number of triangular faces: if the line cuts an existing triangle, one of the resulting two pieces is again a triangle. If it were true more strongly that adding a line always increased the number of triangles, then a similar induction would prove Roberts's theorem, but it is not true. There exist arrangements for which, after adding a line, the number of triangles remains unchanged.^[1]

Instead, Belov uses the following argument. If the ${\displaystyle n}$ given lines are all moved without changing their slopes, their new positions can be described by a system of ${\displaystyle n}$ real numbers, the offsets of each line from its original position. For each triangular face, there is a linear equation on the offsets of its three lines that, if satisfied, causes the face to retain its original area. If there could be fewer than ${\displaystyle n-2}$ triangles, then (because there would be more variables than equations constraining them) it would be possible to fix two of the lines in place and find a simultaneous linear motion of all remaining lines, keeping their slopes fixed, that preserves all of the triangle areas. Such a motion must pass through arrangements that are not simple, for instance when one of the moving lines passes over the crossing point of the two fixed lines. At the time when the moving lines first form a non-simple arrangement, three or more lines meet at a point. Just before these lines meet, by the weaker version of the theorem, the subset of lines that meet would have a triangular face, ${\displaystyle \Delta }$. Because this meeting is the first time the arrangement becomes non-simple, it cannot have changed its combinatorial structure from the original arrangement, which must therefore also contain the same triangle ${\displaystyle \Delta }$. At the time when the lines defining triangle ${\displaystyle \Delta }$ meet, its area becomes zero, but this contradicts the invariance of the areas of the triangles in the initial arrangement. The contradiction shows the impossibility of the assumption that there are fewer than ${\displaystyle n-2}$ triangles.^[1][6]

Whereas Roberts's theorem concerns the fewest possible triangles made by a given number of lines, the related Kobon triangle problem concerns the largest number possible.^[3] The two problems differ already for ${\displaystyle n=5}$, where Roberts's theorem guarantees that three triangles will exist, but the solution to the Kobon triangle problem has five triangles.^[1]

Despite the fact that these seem to be opposite to each other, the Kobon triangle problem makes use of sub-arrangements of lines with minimal triangles. Under the right conditions, for a simple arrangement with ${\displaystyle n}$ lines and maximal triangle count, replacing one line by an arrangement of ${\displaystyle n}$ lines with minimal triangle count creates an arrangement with ${\displaystyle 2n-1}$ lines and maximal triangle count.^[7][8]

Roberts's theorem can be generalized from simple line arrangements to some non-simple arrangements, to arrangements in the projective plane rather than in the Euclidean plane, and to arrangements of hyperplanes in higher-dimensional spaces.^[5] Beyond line arrangements, the same bound as Roberts's theorem holds for arrangements of pseudolines.^[9]