Let @math be a partially ordered set. We prove that if @math is sufficiently large, then there exists a packing @math of copies of @math in the Boolean lattice @math that covers almost every element of @math : @math might not cover the minimum and maximum of @math , and at most @math additional points due to divisibility. In particular, if @math divides @math , then the truncated Boolean lattice @math can be partitioned into copies of @math . This confirms a conjecture of Lonc from 1991.
Recently, problems in which the goal is to partition certain product structures, such as @math @cite_14 , the graph of the hypercube @cite_2 @cite_13 , the Boolean lattice @cite_0 @cite_3 , graph powers of cycles @cite_5 , into copies the definition of copy varies according to the structure we are interested in of some set gained increased interest due to a remarkable result of Gruslys, Leader and Tan @cite_14 . Let us briefly outline their main lemma. While this lemma proved to be powerful in partitioning problems, we were unable to adapt it for "almost partitioning" problems. That is, for problems in which there exists no desired partition for some obvious reason (for example divisibility is not satisfied), but we are still hoping to find a packing that covers almost every element. See @cite_13 @cite_3 for such results. Therefore, in this paper we develop our own packing method to deal with Theorem and Theorem , which exploits the flexibility of copies of posets, and we do not utilize the previously described lemma in any way.
@cite_14 @cite_3 @cite_0 @cite_2 @cite_5 @cite_13
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Let @math be a partially ordered set. If the Boolean lattice @math can be partitioned into copies of @math for some positive integer @math , then @math must satisfy the following two trivial conditions: (1) the size of @math is a power of @math , (2) @math has a unique maximal and minimal element. Resolving a conjecture of Lonc, it was shown by Gruslys, Leader and Tomon that these conditions are sufficient as well. In this paper, we show that if @math only satisfies condition (2), we can still almost partition @math into copies of @math . We prove that if @math has a unique maximal and minimal element, then there exists a constant @math such that all but at most @math elements of @math can be covered by disjoint copies of @math . Let @math be a poset of size @math that has a greatest and a least element. We prove that, for sufficiently large @math , the Boolean lattice @math can be partitioned into copies of @math . This resolves a conjecture of Lonc. Let @math be an induced subgraph of the hypercube @math for some @math . We show that if @math is a power of @math then, for sufficiciently large @math , the vertex set of @math can be partitioned into induced copies of @math . This answers a question of Offner. In fact, we prove a stronger statement: if @math is a subset of @math for some @math and if @math is a power of @math , then, for sufficiently large @math , @math can be partitioned into isometric copies of @math . Let @math be an induced subgraph of the torus @math . We show that when @math is even and @math divides some power of @math , then for sufficiently large @math the torus @math has a perfect vertex-packing with induced copies of @math . On the other hand, disproving a conjecture of Gruslys, we show that when @math is odd and not a prime power, then there exists @math such that @math divides some power of @math , but there is no @math such that @math has a perfect vertex-packing with copies of @math . We also disprove a conjecture of Gruslys, Leader and Tan by exhibiting a subgraph @math of the @math -dimensional hypercube @math , such that there is no @math for which @math has a perfect edge-packing with copies of @math . Let @math be an induced subgraph of the hypercube @math , for some @math . We show that for some @math , the vertices of @math can be partitioned into induced copies of @math and a remainder of at most @math vertices. We also show that the error term cannot be replaced by anything smaller than @math .