On the self crossing six sided figure problem
The author defines an axis parallel hexagon to be, essentially, a six-sided polygon with sides parallel to the axes in which self-crossings of sides do not count as vertices. The main result states that there is a constant \(C>0\) such that, for any \(\mu>0\), if \(E\) is a subset of the unit square that does not contain the vertices of any axis parallel rectangle with area greater than \(\mu^2\) and does not contain the vertices of any axis parallel hexagon with area greater than \(\mu^2\) then \(|E|\leq C\mu\). The theorem partially answers a combinatorial question posed by \textit{A. Carbery, M. Christ} and \textit{J. Wright} [J. Am. Math. Soc. 12, No. 4, 981-1015 (1999)] in relation to the question of the size of the sublevel sets of a highly oscillatory function. The area estimate follows, in turn, from a jigsaw puzzle lemma governing the number of subsets of the square possible under certain conditions on the overlaps of their sections.
scientific article; zbMATH DE number 1348673