Perfect Square Packings

Anchored Rectangle and Square Packings

For points p1, . . . , pn in the unit square [0, 1] , an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r1, . . . , rn ⊆ [0, 1] such that point pi is a corner of the rectangle ri (that is, ri is anchored at pi) for i = 1, . . . , n. We show that for every set of n points in [0, 1], there is an anchored rectangle packing of area at least 7/12 − O(1/n), and...

Square-free perfect graphs

Perfect Packings in Quasirandom Hypergraphs II

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F -packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ≥ 4 and 0 < p < 1. Suppose that H is an n-vertex triple system with r|n and the followin...

Exhaustive approaches to 2D rectangular perfect packings

In this paper, we consider the two-dimensional rectangular strip packing problem, in the case where there is a perfect packing; that is, there is no wasted space. One can think of the problem as a jigsaw puzzle with oriented rectangular pieces. Although this comprises a quite special case for strip packing, we have found it useful as a subroutine in related work. We demonstrate a simple pruning...

Perfect packings in quasirandom hypergraphs I

Let k ≥ 2 and F be a linear k-uniform hypergraph with v vertices. We prove that if n is sufficiently large and v|n, then every quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum degree Ω(nk−1) admits a perfect F -packing. The case k = 2 follows immediately from the blowup lemma of Komlós, Sárközy, and Szemerédi. We also prove positive results for some nonlinea...