This page is under construction. For a link to the previous page detailing the meaning of the universe, press

hoja de la verdad At the moment, this page will serve as a source of information on the Carbery rectangle problem. The simplest form of Carbery's problem is this. Let E be a measurable set which does not contain the vertices of any rectangle with sides parallel to the axes and area greater than a^2. The goal is to show that there is a universal constant C so that the measure of E, |E| <= Ca. With an extra factor of a square root of a logarithm of 1/a, the estimate can be proved trivially. To play a video game which demonstrates this problem on a 30 by 30 grid forbidding rectangles with area greater than 30

press here (Note: This requires a java-enabled browser. Please also note that the undo button does not work with a stack. Rather, it undoes the last move made if it has led to the end of the game or at any other time. In any situation other than the end of the game, if you wish to erase a marked square - simply press it again and hit undo.) The original problem arose from the study of the size of the sublevel sets of a function f on the unit square which has everywhere large mixed second partial. This realization comes from studying integrals of the one form w= (df/dx) dx - (df/dy) dy on the boundary of the rectangle in question and applying Green's theorem. This reasoning also permits us to exclude other polygons with sides parallel to the axis when studying the sublevel set question. If we were allowed to exclude not only axis parallel rectangles and hexagons but also self-crossing axis parallel hexagons, then we would get the sharp bound on the measure of E. For an AMS-TeX file on this, press below.

self-crossing These problems are formally related to questions in extremal graph theory, which may be stated as questions about matrices of 1's and 0's. It is well known, that an N by N matrix of 1s and 0s not containing a full 2 by 2 submatrix has fewer than 2N^{3/2} 1's and that the power of N in this estimate is sharp. If in addition we forbid any 3 by 3 submatrices with 6 1's, we get a power lower than {3/2}. This is essentially the idea of the TeX file above. To play a game about 30 by 30 matrices with full 2 by 2 matrices forbidden and 3 by 3 matrices with 6 1's forbidden

press here If, however, we only forbid those 3 by 3 matrices which give rise to non-self-crossing hexagons, it is not yet known whether one has a bound with a power less than 3/2. If it were known, this would give sharp estimates for the mixed partial problem. To play a game demonstrating this open problem,

press here I would really like people to play this last game. The first version of the applet had a bug in it. The highest reported score was 148, but it means nothing. On the improved version, the highest score I have seen thus far is 109. Probably more is possible. If you do better, please write at nets@math.uic.edu and let me know how. Thanks to all!