# Adrian Dumitrescu

Art could come from anywhere. One just wants to be careful and not
overlook it.

It is like when playing tennis: you have to show up at the game if you
want to win.

T. Rado conjectured in 1928 that if F is a finite set of
axis-parallel

squares in the plane, then there exists an independent subset

I of pairwise disjoint squares, such that I covers at least 1/4 of
the area covered by F.

This conjecture was disproved in 1973 by Ajtai. The construction
we illustrate here

(2008, by Bereg, Dumitrescu and Jiang) is a refinement of Ajtai's
construction

and yields the current record upper bound, 1/4 - 1/384.

What you see is a composite tile made of larger squares of
size10,

smaller ones of sizes 1 and 2, and holes (the coloured parts).

By replicating this composite tile one gets a plane tiling where
every independent set

covers a fraction of at most 1/4 - 1/384 of the total union area.