If we order the points cyclically in their order as vertices of the convex hull then the the (2,2) Radom partition is described by interlacing indices.

]]>Staircase convexity itself may have been introduced in the Bukh et al paper that you mention, but the correspondence between the “lines” defining it (the boundaries of lower-left and lower-right quadrants) and straight lines can already be found in Middendorf, Matthias, and Pfeiffer, Frank. 1992. The max clique problem in classes of string-graphs. Discrete Math., 108(1–3), 365–372.

There’s also a chapter about the same staircase geometry in my recent book Forbidden Configurations in Discrete Geometry, which includes a polynomial time algorithm for recognizing (general position) order types of configurations of points on stretched grids.

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